PEI Math 521K flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Least upper bound property

Every nonempty subset of R that is bounded above has a least upper bound in R. This completeness axiom distinguishes R from Q and underlies monotone convergence and the IVT.

Archimedean property

For every real x there exists a natural number n with n>x. Consequently 1/n can be made smaller than any positive ε, so inf{1/n : n∈N}=0.

Negation of a universal statement

The negation of 'for all x, P(x)' is 'there exists x such that not P(x)'. A single counterexample therefore disproves a universal claim.

Supremum epsilon characterization

A number s is the supremum of A iff s is an upper bound and for every ε>0 some a in A satisfies a>s−ε. The dual statement characterizes the infimum.

Monotone convergence theorem

Every sequence in R that is monotone and bounded converges. An increasing bounded sequence converges to the supremum of its terms.

Bolzano–Weierstrass theorem

Every bounded sequence of real numbers has a convergent subsequence. In R^n the same holds for bounded sequences of vectors.

Cauchy criterion for series

The series ∑a_n converges iff its partial sums form a Cauchy sequence: for every ε>0 there is N with |∑_{k=m+1}^{n} a_k|<ε whenever n>m≥N.

Ratio and root tests

For a series with nonzero terms, limsup|a_{n+1}/a_n|<1 or limsup|a_n|^{1/n}<1 implies absolute convergence; a value greater than 1 implies divergence.

Alternating series test

If b_n decreases monotonically to 0, then ∑(−1)^n b_n converges, and the remainder after N terms is bounded by b_{N+1} in absolute value.

Heine–Cantor theorem

Every continuous function on a compact metric space, in particular on a closed bounded interval, is uniformly continuous. Compactness supplies a δ independent of the point.

Extreme value theorem

A continuous real-valued function on a nonempty compact set attains both a maximum and a minimum. Closed and bounded intervals in R satisfy the hypothesis.

Uniform continuity definition

For every ε>0 there is δ>0 such that |x−y|<δ implies |f(x)−f(y)|<ε for all x,y in the domain; δ depends only on ε, not on the point.

Sequential criterion for continuity

f is continuous at c iff every sequence x_n→c satisfies f(x_n)→f(c). Equivalently, preimages of open sets are open.

Rolle's theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then some c in (a,b) satisfies f'(c)=0.

Mean value theorem

If f is continuous on [a,b] and differentiable on (a,b), then some c satisfies f'(c)=(f(b)−f(a))/(b−a), equating instantaneous and average rates of change.

Fermat interior extremum theorem

If f has a local maximum or minimum at an interior point c and f'(c) exists, then f'(c)=0. Extrema on closed intervals may also occur at endpoints.

Taylor's theorem with remainder

If f is n+1 times differentiable, then f(x)=Σ_{k=0}^{n} f^{(k)}(a)(x−a)^k/k! + f^{(n+1)}(ξ)(x−a)^{n+1}/(n+1)! for some ξ between a and x.

Riemann integrability criterion

A bounded f on [a,b] is Riemann integrable iff for every ε>0 there is a partition P with U(f,P)−L(f,P)<ε, where U and L are upper and lower Darboux sums.

Fundamental theorem of calculus

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).

p-integral convergence

∫_1^∞ x^{−p} dx converges iff p>1, and ∫_0^1 x^{−p} dx converges iff p<1. These are the standard comparison models for improper integrals.

Lebesgue integrability criterion

A bounded function on a closed bounded interval is Riemann integrable iff its set of discontinuities has measure zero. Continuous functions satisfy this automatically.

Uniform convergence definition

f_n→f uniformly on E iff for every ε>0 there is N such that |f_n(x)−f(x)|<ε for all n≥N and all x in E; equivalently sup_E|f_n−f|→0.

Weierstrass M-test

If |f_n(x)|≤M_n for all x in E and the numerical series ∑M_n converges, then ∑f_n converges uniformly and absolutely on E.

Dini's theorem

If a sequence of continuous functions increases or decreases monotonically to a continuous limit on a compact set, then the convergence is uniform.

Power series radius of convergence

For ∑a_n(x−c)^n the radius is R=1/limsup|a_n|^{1/n}. The series converges absolutely for |x−c|<R and uniformly on compact subintervals.

Compactness via open covers

A set is compact if every open cover admits a finite subcover. In metric spaces this is equivalent to every sequence having a convergent subsequence with limit in the set.

Heine–Borel theorem

A subset of R^n is compact iff it is closed and bounded. In general metric spaces compact sets are closed and bounded, but the converse can fail.

Total boundedness

For every ε>0 finitely many ε-balls cover the set. A metric space is compact iff it is complete and totally bounded.

Total derivative and Jacobian

f is differentiable at a if f(a+h)=f(a)+Df(a)h+o(|h|) for a linear map Df(a), whose matrix of partial derivatives is the Jacobian.

Inverse function theorem

If f is C¹ near a and Df(a) is invertible, then f is locally a C¹ bijection and D(f⁻¹)(f(a)) is the inverse matrix of Df(a).

Implicit function theorem

If F(a,b)=0, F is C¹, and the partial Jacobian D_yF(a,b) is invertible, then F(x,y)=0 defines y=g(x) locally with Dg=−(D_yF)⁻¹D_xF.

Proof by contrapositive

To prove P implies Q, prove the logically equivalent statement that not Q implies not P. This differs from proving the converse Q implies P, which is not equivalent.