PEI Math 521K study plan

Write out the field and order axioms for R from memory, then prove three consequences (uniqueness of 0 and 1, cancellation laws, sign rules for products of negatives) without consulting notes.

Prove by induction that 1+2+...+n = n(n+1)/2 and that 2^n > n for all n ≥ 1, stating the base case and inductive step explicitly each time.

Translate five English statements into quantified logic with correct quantifier order, and negate each one mechanically (swap ∀/∃, negate the predicate).

Distinguish valid proof patterns (direct, contrapositive, contradiction, induction) by matching each of eight short claims to the technique that actually works, and explain why the others fail.

Prove the Archimedean property and the density of Q in R using only completeness (supremum) as a starting point.

Prove that every nonempty subset of R bounded above has a least upper bound, and use it to derive the nested interval property and the Bolzano–Weierstrass theorem for sequences.

State the ε-N definition of convergence and use it to prove limits of at least four sequences from scratch, including one where you must choose N as an explicit function of ε.

Prove that convergent sequences are bounded and that limits are unique, writing the triangle-inequality estimates explicitly.

Classify monotone sequences using the monotone convergence theorem, and apply it to a recursively defined sequence to prove convergence and identify the limit.

Define limsup and liminf via ε-characterizations and prove that lim a_n exists iff limsup a_n = liminf a_n.

Prove the Cauchy criterion for sequences and give an example of a Cauchy sequence in a non-complete setting to show why completeness matters.

Apply the comparison, ratio, root, and integral tests to at least eight series, justifying convergence or divergence in each case with the hypotheses checked.

Prove absolute convergence implies convergence, and construct a conditionally convergent series whose terms can be rearranged to a different sum.

Prove the alternating series test and estimate the remainder for a specific series with an explicit error bound.

Compute limits of sequences using standard limits, Stolz–Cesàro or l'Hôpital-style reductions where legitimate, and justify each substitution.

Write the ε-δ definition of continuity and prove continuity of polynomials, rational functions off their zeros, and compositions from the definitions.

Distinguish pointwise from uniform continuity with a proof that f(x) = x² is not uniformly continuous on R but is on [0, 1], using the sequential criterion.

Prove that continuous functions on compact intervals attain their extrema and are uniformly continuous there (Heine–Cantor), citing compactness correctly.

Apply the intermediate value theorem to prove existence of roots, and use the extreme value theorem to justify optimization claims.

Identify and repair three common logical errors in continuity proofs (choosing δ after ε-dependent quantities, confusing pointwise with uniform, ignoring domain restrictions).

Prove differentiability implies continuity, and give an example of a continuous nowhere-differentiable-in-a-point function to separate the notions.

Derive the product, quotient, and chain rules from the limit definition, stating each hypothesis.

State and prove Rolle's theorem, then derive the mean value theorem and use it to prove that a function with zero derivative on an interval is constant.

Apply the MVT and Cauchy MVT to prove inequalities and to establish l'Hôpital's rule for the 0/0 case.

Prove Taylor's theorem with Lagrange remainder and use it to bound approximation error for a specific function.

Prove integrability of continuous functions on closed intervals using uniform continuity and Riemann sums, and state the Darboux criterion for integrability.

Prove the fundamental theorem of calculus in both parts, carefully tracking which hypotheses (continuity, integrability) are needed for each.

Evaluate improper integrals of both types, deciding convergence with comparison and limit tests, and prove convergence or divergence for at least five examples.

Distinguish pointwise from uniform convergence of function series, and prove the Weierstrass M-test and use it on a power series.

Prove that uniform convergence preserves continuity and permits term-by-term integration, and give a counterexample showing pointwise convergence does not.

Verify the metric space axioms for R^n with Euclidean, taxicab, and sup metrics, and prove that the three are equivalent for convergence.

Prove that compactness, sequential compactness, and closed-and-bounded are equivalent in R^n (Heine–Borel), and identify which implications fail in general metric spaces.

Determine openness, closedness, closure, interior, and boundary for at least six concrete subsets of R², justifying each claim with a proof or counterexample.

Prove that continuous images of compact sets are compact and that continuous real-valued functions on compact sets attain extrema.

Use the Baire category or nested compact set argument to prove a nonempty intersection result, and explain where completeness is used.

Compute partial derivatives, gradients, Jacobians, and directional derivatives for multivariable functions, and verify differentiability via the linear approximation definition rather than partials alone.

Prove the chain rule in several variables and use it to compute derivatives of compositions.

State and apply the inverse function theorem, checking the nonvanishing Jacobian hypothesis, and compute the derivative of an inverse map.

State and apply the implicit function theorem to solve for one variable in terms of others, verifying hypotheses, and compute implicit derivatives.

Assemble a timed 3-hour mock exam covering all nine domains and grade it against a rubric, logging every lost point by domain.