PEI Math 521K cheat sheet

Order axioms: R is a field with a total order compatible with + and ·; exactly one of a<b, a=b, b<a holds, and a<b, b<c imply a<c.

Completeness (least upper bound property): every nonempty subset of R bounded above has a supremum in R; this is what distinguishes R from Q.

Archimedean property: for any x in R there is n in N with n>x; consequence: for every ε>0 there is n with 1/n<ε, so inf{1/n}=0.

Density of Q and of irrationals: between any two reals a<b lie a rational and an irrational; proof uses Archimedes plus the irrationality of √2.

Supremum characterization: s=sup A iff s is an upper bound and for every ε>0 some a in A satisfies a>s−ε; dually for inf.

Logic quantifiers: ¬∀x P(x) ≡ ∃x ¬P(x) and ¬∃x P(x) ≡ ∀x ¬P(x); order of quantifiers matters - ∀x∃y P ≠ ∃y∀x P in general.

Proof templates: to prove A⊆B take arbitrary x∈A and derive x∈B; to prove A=B show both inclusions; to disprove ∀x P give one counterexample.

Triangle and absolute value: |a+b| ≤ |a|+|b|, | |a|−|b| | ≤ |a−b|; |x|<ε iff −ε<x<ε.

Induction and well-ordering: N is well-ordered, so any nonempty subset has a least element; strong induction on n assumes the statement for all k<n.

Convergence: a_n→L iff for every ε>0 there is N with |a_n−L|<ε for all n≥N; equivalently every ε-neighborhood of L contains all but finitely many terms.

A convergent sequence is bounded and has a unique limit; every subsequence of a convergent sequence converges to the same limit.

Monotone convergence theorem: a bounded monotone sequence in R converges; sup of an increasing bounded sequence equals its limit.

Bolzano–Weierstrass: every bounded sequence in R (or in R^n) has a convergent subsequence; Cauchy sequences converge in R (completeness).

Limit algebra: sums, products, quotients (denominator limit nonzero) and continuous images of convergent sequences behave as expected; squeeze theorem: a_n≤b_n≤c_n with a_n,c_n→L gives b_n→L.

Series: ∑a_n converges iff partial sums S_N are Cauchy; if ∑a_n converges then a_n→0 (converse false, harmonic series diverges).

Nonnegative series tests: comparison, limit comparison, ratio (lim|a_{n+1}/a_n|<1 converges, >1 diverges), root (limsup|a_n|^{1/n}<1 converges), integral test for decreasing positive f.

Alternating series test: if b_n decreases to 0 then ∑(−1)^n b_n converges; absolute convergence implies convergence, but conditional convergence (∑(−1)^n/n) does not.

limsup and liminf: limsup a_n = lim_{n} sup_{k≥n} a_k; a_n converges iff liminf=limsup is finite; ratio/root tests use limsup to handle all cases.

Continuity at c: for every ε>0 there is δ>0 such that |x−c|<δ implies |f(x)−f(c)|<ε; δ may depend on both ε and c.

Sequential criterion: f is continuous at c iff x_n→c implies f(x_n)→f(c); equivalently preimages of open sets are open (relative to the domain).

Uniform continuity: for every ε>0 there is δ>0 depending only on ε such that |x−y|<δ implies |f(x)−f(y)|<ε for all x,y in the domain.

Heine–Cantor: a continuous function on a compact set (closed and bounded in R^n) is uniformly continuous; compactness defeats the failure of the naive δ.

Extreme value theorem: continuous f on compact K attains max and min; continuous image of a compact set is compact; continuous image of a connected set is an interval (IVT).

Continuous on [a,b] implies bounded and integrable; continuity on a closed bounded interval is not uniform only when the domain is non-compact, e.g. f(x)=1/x on (0,1).

Lipschitz condition |f(x)−f(y)|≤M|x−y| implies uniform continuity; √x on [0,∞) is uniformly continuous but not Lipschitz.

Composition and algebra: sums, products, quotients (denominator nonzero) and compositions of continuous functions are continuous; uniform continuity is preserved under composition with a uniformly continuous outer function.

Discontinuity classification: f has a removable, jump, or essential discontinuity at c according to the behavior of one-sided limits; monotone functions have at most countably many jump discontinuities.

Derivative: f'(c)=lim_{h→0}(f(c+h)−f(c))/h; differentiability at c implies continuity at c, but not conversely (|x| at 0).

Rules: (fg)'=f'g+fg', (f/g)'=(f'g−fg')/g², chain rule (f∘g)'=f'(g(x))g'(x); derivative of inverse (f⁻¹)'(y)=1/f'(x) when f'(x)≠0.

Fermat: if f has a local extremum at an interior point c and f'(c) exists, then f'(c)=0; critical points plus endpoints are the only candidates for extrema.

Rolle: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) implies some c with f'(c)=0.

Mean value theorem: f continuous on [a,b], differentiable on (a,b) implies some c with f'(c)=(f(b)−f(a))/(b−a); corollary: f'≡0 implies f constant.

Monotonicity test: f'>0 on an interval implies strictly increasing; f'≥0 implies nondecreasing; sign of f' determines local behavior.

L'Hôpital and Taylor: for 0/0 or ∞/∞ forms apply lim f'/g' when the latter exists; Taylor's theorem gives f(x)=Σ_{k≤n} f^{(k)}(a)(x−a)^k/k! + R_n with Lagrange remainder f^{(n+1)}(ξ)(x−a)^{n+1}/(n+1)!.

Darboux property: derivatives have the intermediate value property even when discontinuous; Cauchy MVT generalizes to (f(b)−f(a))g'(c)=(g(b)−g(a))f'(c).

Riemann integrability: f on [a,b] is integrable iff for every ε>0 there is a partition P with U(f,P)−L(f,P)<ε; equivalently inf over partitions of U equals sup of L.

Continuous functions on [a,b] and monotone functions on [a,b] are Riemann integrable; a bounded function with finitely many discontinuities is integrable.

Lebesgue criterion: bounded f on [a,b] is Riemann integrable iff its set of discontinuities has measure zero.

Properties: linearity, additivity over adjacent intervals, |∫f|≤∫|f|, and if m≤f≤M then m(b−a)≤∫f≤M(b−a).

Fundamental theorem: if f is continuous on [a,b] then F(x)=∫_a^x f is differentiable with F'=f; if F'=f and f is integrable then ∫_a^b f=F(b)−F(a).

Improper integrals: ∫a^∞ f = lim{R→∞}∫_a^R f; convergence requires the limit to exist; ∫_1^∞ 1/x^p converges iff p>1, and ∫_0^1 1/x^p converges iff p<1.

Comparison and absolute convergence: if |f|≤g and ∫g converges then ∫f converges; absolute convergence implies convergence, conditional examples exist via cancellation.

Integration by parts and substitution: ∫_a^b u dv = uv|_a^b − ∫_a^b v du, and substitution requires the change of variable to be C¹ on the interval.

Pointwise convergence: f_n→f on E iff for each x and each ε>0 there is N(x,ε) with |f_n(x)−f(x)|<ε for n≥N.

Uniform convergence: for every ε>0 there is N(ε) with |f_n(x)−f(x)|<ε for all n≥N and all x∈E; equivalently sup_{x∈E}|f_n(x)−f(x)|→0.

Weierstrass M-test: if |f_n(x)|≤M_n for all x and ∑M_n converges, then ∑f_n converges uniformly (and absolutely).

Uniform limit of continuous functions is continuous; pointwise limit need not be (f_n(x)=x^n on [0,1] has discontinuous limit).

Interchange theorems: uniform convergence permits lim∫f_n=∫lim f_n on bounded intervals and, with continuous f_n' converging uniformly plus one convergent point, lim f_n'=(lim f_n)'.

Dini's theorem: if f_n is continuous, monotone, and converges pointwise to continuous f on a compact set, then the convergence is uniform.

Power series: ∑a_n(x−c)^n converges uniformly on compact subsets inside its radius R=1/limsup|a_n|^{1/n}; term-by-term differentiation and integration are valid for |x−c|<R.

Uniform Cauchy criterion: f_n converges uniformly iff for every ε>0 there is N with sup_x|f_n(x)−f_m(x)|<ε for all m,n≥N; this is the standard tool when the limit is unknown.

Metric axioms: d(x,y)≥0 with equality iff x=y, d(x,y)=d(y,x), and d(x,z)≤d(x,y)+d(y,z); open ball B(x,r)={y:d(x,y)<r}.

Open sets are unions of open balls; closed sets are complements of open sets; arbitrary unions and finite intersections of open sets are open (dual for closed).

Closure, interior, boundary: x∈cl A iff every ball around x meets A; x∈int A iff some ball lies in A; ∂A=cl A ∩ cl(A^c).

Continuity in metric spaces: f is continuous iff preimages of open sets are open, iff preimages of closed sets are closed, iff x_n→x implies f(x_n)→f(x).

Compactness: every open cover has a finite subcover; sequentially compact means every sequence has a convergent subsequence; in metric spaces these are equivalent.

Heine–Borel: a subset of R^n is compact iff it is closed and bounded; in general metric spaces compact implies closed and bounded but not conversely.

Compact sets are closed and bounded; continuous images of compact sets are compact; a continuous real function on a compact set attains its extrema.

Completeness and total boundedness: a metric space is compact iff it is complete and totally bounded; R^n is complete under the Euclidean metric.

Connectedness: a subset of R is connected iff it is an interval; continuous images of connected sets are connected; R^n is path connected.

Total derivative: f:R^n→R^m is differentiable at a if there is a linear map Df(a) with f(a+h)=f(a)+Df(a)h+o(|h|); Df(a) is the Jacobian matrix of partials when it exists.

Existence of partial derivatives does not imply differentiability; a sufficient condition is that all partials exist and are continuous near a (C¹ implies differentiable).

Chain rule: D(f∘g)(a)=Df(g(a))·Dg(a) as matrix multiplication; directional derivative D_v f(a)=Df(a)v.

Inverse function theorem: if f:R^n→R^n is C¹ near a and Df(a) is invertible, then f is locally a C¹ bijection with D(f⁻¹)(f(a))=(Df(a))⁻¹.

Implicit function theorem: if F(x,y)=0, F is C¹ near (a,b), F(a,b)=0, and the partial Jacobian D_yF(a,b) is invertible, then y=g(x) locally with g'(x)=−(D_yF)⁻¹D_xF.

Second derivative test: classify critical points of f:R^n→R by the Hessian - positive definite gives a local minimum, negative definite a maximum, indefinite a saddle.

Taylor in several variables: f(a+h)=f(a)+∇f(a)·h + ½ h^T H_f(a) h + o(|h|²) when f is C².

Lagrange multipliers: extrema of f subject to g=0 occur where ∇f=λ∇g provided ∇g≠0; the constraint set must be a regular level surface.

Direct proof: assume the hypotheses, unpack definitions, and derive the conclusion; every step must cite a definition, axiom, or previously proved theorem.

Contrapositive: prove ¬Q→¬P instead of P→Q; this is logically equivalent, whereas the converse Q→P is not.

Contradiction: assume the negation of the goal and derive a false statement; useful for irrationality of √2 and for uniqueness proofs.

Counterexample: a single instance satisfying the hypotheses but failing the conclusion disproves a universal statement; existence claims require an explicit construction.

Quantifier discipline: write 'for every ε>0 there exists δ>0 such that ...' explicitly; a δ depending on x is not uniform, and swapping ∀∃ invalidates the claim.

Definitions are biconditionals in practice: to prove continuity, convergence, or compactness, verify the defining condition, not a remembered paraphrase.

ε-N and ε-δ estimates: bound |a_n−L| or |f(x)−f(c)| by an explicit expression in ε, choosing N or δ last; triangle inequality is the standard tool.

Induction: base case, inductive hypothesis for all k≤n (strong) or for n (weak), then the step; well-ordering supports minimal-counterexample arguments.

Proof by cases and without loss of generality: split on a genuine dichotomy (x≥0 or x<0, finite or infinite); justify any symmetry claim before invoking WLOG.

Dedekind cuts and Cauchy completion both construct R from Q; both yield the same complete ordered field up to unique order-isomorphism.

Nested interval property: a nested sequence of closed bounded intervals [a_n,b_n] with b_n−a_n→0 has exactly one point in the intersection.

Bernoulli and binomial inequalities: (1+x)^n ≥ 1+nx for x≥−1 and n∈N; used to show (1+1/n)^n increases and is bounded.

Countability: Q is countable, R is uncountable (Cantor diagonal argument); a countable union of countable sets is countable.

Logical equivalence: P→Q ≡ ¬P∨Q ≡ ¬Q→¬P; negation of 'for all x there exists y with P' is 'there exists x such that for all y, not P'.

Sup and inf arithmetic: sup(A+B)=sup A+sup B when both are finite; sup(cA)=c·sup A for c>0 and c·inf A for c<0.

Rational roots and irrationality: if p is prime then √p is irrational; proof assumes √p=m/n in lowest terms and derives p|m and p|n.

Standard limits: (1+1/n)^n→e, n^{1/n}→1, (1+c/n)^n→e^c, and a^n→0 for |a|<1; polynomial growth dominates logarithms and is dominated by exponentials.

Cesàro and Stolz–Cesàro: if a_n→L then the averages (a_1+...+a_n)/n→L; the converse fails (alternating sequence).

Cauchy condensation: for a nonincreasing positive a_n, ∑a_n converges iff ∑2^k a_{2^k} converges; proves divergence of ∑1/n and convergence of ∑1/(n(log n)^p) for p>1.

Rearrangement: absolutely convergent series may be reordered without changing the sum; conditionally convergent series can be rearranged to any prescribed sum (Riemann).

Limit points and subsequences: L is a limit point of (a_n) iff some subsequence converges to L; limsup is the largest such L and liminf the smallest.

Contractive sequences: if |a_{n+1}−a_n|≤c|a_n−a_{n−1}| with 0<c<1, the sequence is Cauchy and converges.

Comparison at infinity: if a_n/b_n→c with 0<c<∞ then ∑a_n and ∑b_n converge or diverge together (limit comparison test).

One-sided continuity: f is continuous at an interior point c iff both one-sided limits exist and equal f(c); on a closed interval continuity at endpoints is one-sided.

Intermediate value theorem: continuous f on [a,b] with f(a)<y<f(b) attains y at some c in (a,b); used for root existence and fixed points.

Fixed point: a continuous map of [a,b] into itself has a fixed point; a contraction with constant c<1 has a unique fixed point (Banach).

Uniform continuity fails on unbounded domains for functions with unbounded derivative, e.g. f(x)=x² on R; it holds for f(x)=sin(x²)? No - sin(x²) is uniformly continuous on R while x² is not.

Continuity and monotonicity: a continuous strictly monotone function on an interval has a continuous inverse on its image.

Oscillation: f is continuous at c iff the oscillation osc(f,c)=inf over neighborhoods of the diameter of f(U) is zero; the set of discontinuities is a countable union of closed sets.

Extension: a uniformly continuous function on a dense subset of a complete metric space extends uniquely to a uniformly continuous function on the whole space.

Cauchy MVT: for f,g continuous on [a,b] and differentiable inside, there is c with (f(b)−f(a))g'(c)=(g(b)−g(a))f'(c); the ordinary MVT is g(x)=x.

Convexity: f''≥0 on an interval iff f is convex; convex functions lie above their tangent lines and satisfy f((x+y)/2)≤(f(x)+f(y))/2.

Inverse function (one variable): if f' is continuous and nonzero at a, f is locally invertible and (f⁻¹)'(f(a))=1/f'(a).

Higher derivatives and extrema: if f'(c)=f''(c)=...=f^{(n−1)}(c)=0 and f^{(n)}(c)≠0, then c is a local extremum when n is even and not an extremum when n is odd.

Rolle's theorem is the engine for uniqueness: if two solutions of a differential equation exist, their difference has a zero derivative and hence is constant.

Derivative bounds: |f'(x)|≤M on an interval implies |f(x)−f(y)|≤M|x−y|; the MVT gives the reverse estimate for differentiable functions.

Taylor remainder forms: Lagrange (uses f^{(n+1)}(ξ)) and integral form ∫_a^x f^{(n+1)}(t)(x−t)^n/n! dt when f is C^{n+1}.

Darboux sums: for a partition P, L(f,P)=Σ m_i Δx_i and U(f,P)=Σ M_i Δx_i with m_i, M_i the inf and sup of f on each subinterval; refinement shrinks the gap.

Riemann sums: if f is integrable, any tagged partition with mesh →0 has Riemann sums converging to ∫_a^b f.

Cauchy–Schwarz and Hölder for integrals: (∫fg)²≤(∫f²)(∫g²) on [a,b]; useful for bounding error terms.

Improper integrals with interior singularities: split at the singularity, e.g. ∫0^1 x^{−1/2} dx = lim{a→0+}∫_a^1; both pieces must converge independently.

Absolute vs conditional: ∫_1^∞ sin x / x dx converges conditionally; ∫_1^∞ |sin x|/x dx diverges (comparison with harmonic blocks).

Mean value theorem for integrals: for continuous f on [a,b] there is c with ∫_a^b f = f(c)(b−a); the first mean value theorem for integrals extends this with a weight g≥0.

Integration of uniform limits: if f_n→f uniformly on [a,b] and each f_n is integrable, then f is integrable and ∫f_n→∫f.

Sup norm test: uniform convergence of f_n to f on E is equivalent to ‖f_n−f‖_sup→0; pointwise convergence is only f_n(x)→f(x) for each fixed x.

Uniform convergence preserves boundedness and continuity; it also preserves integrability on bounded intervals and allows swapping limit and integral.

Differentiation is stricter: uniform convergence of f_n' plus convergence of f_n at one point gives (lim f_n)'=lim f_n' on the interval.

Abel and Dirichlet tests for uniform convergence of ∑a_n(x)b_n(x): monotone bounded b_n plus a uniformly bounded or convergent ∑a_n suffice.

Power series facts: radius R=1/limsup|a_n|^{1/n}; convergence is absolute for |x−c|<R, uniform on compact subintervals, and term-by-term operations are valid inside R.

Equicontinuity and Arzelà–Ascoli: a uniformly bounded, equicontinuous sequence of functions on a compact set has a uniformly convergent subsequence.

Stone–Weierstrass: polynomials are uniformly dense in C[a,b]; trigonometric polynomials are dense in continuous 2π-periodic functions.

Equivalent metrics: metrics d and d' inducing the same open sets give the same continuous functions, convergent sequences, and compact sets.

Subspace topology: A⊆X inherits open sets U∩A; a subset of R^n is relatively open in A iff it is the intersection of A with an open set of R^n.

Separation in metric spaces: a metric space is Hausdorff, so limits of sequences are unique and compact subsets are closed.

Cantor intersection theorem: a decreasing sequence of nonempty compact sets has nonempty intersection; in complete spaces use closed sets with diameters →0.

Total boundedness: for every ε>0 finitely many ε-balls cover the set; compact implies totally bounded, and in R^n compactness equals closed plus bounded.

Continuous bijection from compact to Hausdorff is a homeomorphism; this is why inverse continuity often comes free on compact domains.

Baire category: a complete metric space is not a countable union of nowhere dense sets; used to prove existence of continuous nowhere differentiable functions.

Jacobian determinant measures local volume scaling; the total derivative is the best linear approximation, and differentiability implies continuity.

Mixed partials: if f is C² then ∂²f/∂x∂y=∂²f/∂y∂x (Clairaut/Schwarz); the hypothesis of continuity of second partials cannot be dropped.

Mean value inequality: if ‖Df‖≤M on a convex set then ‖f(x)−f(y)‖≤M‖x−y‖; a vector-valued MVT with equality fails, so use the integral form.

Regular value: if Df has full rank at every point of f⁻¹(c), that level set is a C¹ manifold of dimension n−m (preimage theorem).

Implicit differentiation: differentiate F(x,g(x))=0 to get D_xF + D_yF·Dg=0, solving for Dg when D_yF is invertible.

Constrained extrema: solve ∇f=λ∇g together with g=0; check that ∇g≠0 on the constraint, otherwise use a parametrization.

Contraction mapping theorem: a contraction on a complete metric space has a unique fixed point; this proves both inverse and implicit function theorems.

Uniqueness template: suppose two objects satisfy the defining property and show they are equal; for limits, use |L−L'|<ε for all ε>0 to conclude L=L'.

Existence template: construct the object explicitly or invoke a theorem (IVT, EVT, completeness, contraction mapping) and verify its hypotheses one by one.

Negating compound statements: ¬(P∧Q) ≡ ¬P∨¬Q, ¬(P∨Q) ≡ ¬P∧¬Q, and ¬(P→Q) ≡ P∧¬Q; the last is the shape of a counterexample.

Choosing ε: to prove a limit, pick ε first; to prove a bound, choose N=⌈1/ε⌉ or δ=min(1,ε/(2M+1)) so all cases stay under ε.

Monotone and bounded estimates: replace unknown quantities by explicit bounds (|sin x|≤1, |x−c|<δ≤1) before applying the triangle inequality.

Proof by minimal counterexample: assume a counterexample exists, take the least one by well-ordering, and derive a smaller one - equivalent to strong induction.

Rigor check: verify each hypothesis (compactness, continuity, differentiability, nonzero denominator) before applying a theorem, and state where it is used.

To compute sup A: guess the value, prove it is an upper bound, then for each ε>0 exhibit a∈A with a>s−ε.

To show a set is unbounded above: for any M find an element exceeding M; the Archimedean property handles sets like {n−1/n}.

To prove an inequality for all n: try induction, or prove the stronger monotone statement and compare with the limit.

To show a number is irrational: assume a/b in lowest terms, derive a divisibility contradiction, and cite unique factorization.

To negate a convergence statement: write ∃ε>0 such that ∀N ∃n≥N with |a_n−L|≥ε; this is the template for divergence proofs.

To compare cardinalities: an injection A→B plus an injection B→A gives a bijection (Cantor–Bernstein); diagonalization shows R is uncountable.

To find lim a_n: try monotone convergence (show bounded and monotone), squeeze with known limits, or take logarithms for products and powers.

Recursive sequences: show the limit L exists by monotonicity and boundedness, then pass to the limit in a_{n+1}=f(a_n) to solve L=f(L).

To test a series: check a_n→0 first; then try comparison or limit comparison, ratio/root for factorials and powers, integral test for decreasing positive terms, alternating test for sign changes.

To prove divergence: exhibit a divergent minorant (comparison) or show the terms do not tend to zero; for the harmonic series use condensation or block estimates.

To estimate a remainder: for alternating series |R_N|≤b_{N+1}; for the integral test use ∫_{N+1}^∞ f ≤ R_N ≤ ∫_N^∞ f.

To handle limsup: use limsup(a_n+b_n)≤limsup a_n+limsup b_n and limsup(a_n b_n)≤(limsup a_n)(limsup b_n) for nonnegative sequences.

To prove continuity at c: bound |f(x)−f(c)| by an expression in |x−c|, then choose δ; factor differences for polynomials and use |√x−√c|≤|x−c|/√c.

To prove uniform continuity: find a Lipschitz constant, or use compactness via Heine–Cantor, or split the domain into a compact part and a tail where the function is small.

To disprove uniform continuity: produce sequences x_n,y_n with |x_n−y_n|→0 but |f(x_n)−f(y_n)|≥ε₀; for x² use x_n=n, y_n=n+1/n.

To apply the IVT: verify continuity on a closed interval and check the sign change at endpoints; for fixed points consider g(x)=f(x)−x.

To use the EVT: verify the domain is compact (closed and bounded) before asserting a maximum exists; on open intervals the max may be approached but not attained.

To extend a continuous function: check uniform continuity on the dense subset, then invoke the unique extension theorem to the completion.

To prove an inequality f(x)≤g(x): set h=f−g, show h(a)≤0 and h'≤0 on the interval, then apply the MVT.

To count roots: use Rolle's theorem - between two roots of f lies a root of f'; if f' has at most k roots then f has at most k+1.

To evaluate a limit: identify the indeterminate form, then use L'Hôpital, Taylor expansion, or factoring; verify the hypotheses of L'Hôpital before applying it.

To bound a function's growth: apply Taylor's theorem with Lagrange remainder and bound the (n+1)-st derivative on the interval.

To show a function is constant: prove its derivative vanishes on a connected interval, or that it is both nondecreasing and nonincreasing.

To prove differentiability at a point: compute the difference quotient limit directly; to disprove it, exhibit two sequences of h→0 giving different quotients.

To prove integrability from the definition: for each ε>0 construct a partition with U−L<ε; uniform continuity on [a,b] supplies the mesh size.

To show non-integrability: exhibit ε₀>0 such that every partition has U−L≥ε₀; for Dirichlet's function the gap is b−a.

To evaluate an improper integral: write the limit explicitly, antidifferentiate, and take the limit; split at interior singularities before comparing.

To test convergence without evaluating: compare with 1/x^p near infinity or near 0, and use limit comparison with the standard p-integrals.

To justify term-by-term integration: establish uniform convergence on the interval (M-test or Dini) before swapping limit and integral.

To handle a parameter integral: prove continuity or differentiability in the parameter by dominated estimates or uniform convergence of the differentiated integrand.

To find the pointwise limit: fix x and compute lim f_n(x) by ordinary sequence techniques; then check whether sup|f_n−f|→0 for uniformity.

To disprove uniform convergence: find x_n with |f_n(x_n)−f(x_n)|≥ε₀; for x^n on [0,1] take x_n=1−1/n.

To prove uniform convergence of a series: apply the Weierstrass M-test with a convergent numerical majorant, or use Abel/Dirichlet with monotone coefficients.

To justify differentiation of a series: show the differentiated series converges uniformly on compact subintervals and the original series converges at one point.

To compute a power series radius: apply the ratio test to |a_{n+1}/a_n| or the root test to |a_n|^{1/n}, then check endpoints separately.

To show a limit function is continuous: prove uniform convergence of continuous functions, or use Dini's theorem when the convergence is monotone on a compact set.

To show a set is open: for each point produce a radius r>0 with B(x,r) inside the set; for closed, show the complement is open or that limits of sequences stay inside.

To show a set is not compact: exhibit an open cover with no finite subcover, or a sequence with no convergent subsequence inside the set.

To show a set is compact in R^n: prove it is closed and bounded, then invoke Heine–Borel; for general metric spaces prove sequential compactness directly.

To use compactness: extract a finite subcover from a cover by ε-balls, or a finite subcover of neighborhoods supplied by continuity at each point.

To prove a map is a homeomorphism: show it is a continuous bijection from a compact space to a Hausdorff space, then cite the theorem.

To work with closures: x∈cl A iff there is a sequence in A converging to x (in metric spaces); use this to translate topological claims into ε-statements.

To check differentiability at a point: compute the Jacobian, then verify f(a+h)−f(a)−Df(a)h = o(|h|); polar coordinates expose failures like xy/(x²+y²).

To apply the inverse function theorem: compute Df(a), verify its determinant is nonzero, and conclude local invertibility and the derivative formula for f⁻¹.

To apply the implicit function theorem: verify F(a,b)=0 and that D_yF(a,b) is invertible, then compute g'=−(D_yF)⁻¹D_xF.

To classify critical points: compute the Hessian, test definiteness via leading principal minors (Sylvester), and fall back to direct estimates when it is degenerate.

To solve constrained optimization: set up ∇f=λ∇g with the constraint, solve the system, and compare values; check boundary and singular points separately.

To prove a vector-valued estimate: integrate Df along the segment from x to y and bound the operator norm, since the scalar MVT does not hold componentwise with equality.

To prove a set equality: show both inclusions; to prove an implication chain, state each implication explicitly rather than writing a wall of symbols.

To prove uniqueness of a limit or of an inverse: assume two candidates and show their distance is less than every ε>0, hence zero.

To structure an ε-proof: fix ε>0, choose N or δ depending only on ε and the data, then verify the estimate for all indices or points in range.

To refute a converse: produce a specific counterexample satisfying the hypothesis but not the conclusion, and verify both properties explicitly.

To use a theorem: list its hypotheses, check each one against the problem data, and only then apply the conclusion; citing a theorem with an unmet hypothesis is the most common lost point.

To organize a long proof: state the goal, isolate the key estimate or construction, prove it as a lemma, then assemble; mark where each hypothesis is used.