PEI Math 521K common mistakes
PEI Math 521K · Understand the traps and how to avoid them.
Proving a sequence converges by computing a limit with limit laws before establishing that the limit exists, e.g. writing L = lim a_n and solving L = f(L) for a recursively defined sequence without first showing (a_n) is monotone and bounded (or Cauchy).
Limit laws are theorems that presuppose convergence; the algebra feels productive and produces a plausible number, so the missing existence argument is invisible to the writer. Recursive sequences make this especially tempting because the fixed-point equation always has a solution.
What to do instead
Before any limit algebra, write one sentence naming the theorem that gives existence: monotone convergence (prove a_{n+1} - a_n has constant sign by induction and prove boundedness by induction), or Cauchy plus completeness, or contractive/Banach fixed point. Then compute. On scratch work, mark every '=' involving lim with the justification that licenses it.
Confusing pointwise and uniform convergence: checking sup |f_n - f| -> 0 only on a sample of points, or declaring uniform convergence from pointwise convergence plus continuity of each f_n, or claiming the limit of continuous functions is continuous without uniform convergence.
The definitions differ only in the order of quantifiers, and pointwise convergence is what students actually compute. The quantifier swap is easy to state and hard to internalize, so the epsilon-N that must be independent of x never gets written.
What to do instead
Always compute M_n = sup_{x in E} |f_n(x) - f(x)| explicitly (differentiate, or bound by a uniform estimate) and show M_n -> 0. When uniformity fails, produce the specific witness: an epsilon_0 > 0 and a sequence x_n with |f_n(x_n) - f(x_n)| >= epsilon_0. Memorize the standard witnesses (x^n on [0,1], nx(1-x)^2, x/(1+nx)) as templates.
Applying a theorem outside its hypotheses: using the mean value theorem on a function not differentiable on the open interval, invoking the inverse function theorem where the derivative is not invertible, or citing 'f continuous on a compact set' for a set that is only closed or only bounded.
Hypothesis-checking is treated as a formality rather than part of the proof, and exam statements are engineered so that the interesting case is exactly the boundary case where a hypothesis fails (e.g. f(x) = |x|, x^{1/3}, or 1/x on (0,1)).
What to do instead
Adopt a ritual: before naming a theorem, list its hypotheses as bullet points and cite the line of the problem that supplies each one. If a hypothesis is not supplied, either prove it (e.g. show the interval is compact, show f' is continuous and nonzero) or split into cases. Keep a one-page table of theorem hypotheses and review it before the exam.
Treating uniform continuity as ordinary continuity: proving 'for each x there is delta' and calling it uniform, or asserting that a continuous function on a bounded set is uniformly continuous, or that 1/x is uniformly continuous on (0,1).
The epsilon-delta quantifier order (delta must work for all x simultaneously) is not visible in the notation, and most examples in early courses are on compact intervals where the two notions coincide, so the distinction never bites until the exam.
What to do instead
Write the definition with quantifiers in explicit order and, in every proof, exhibit a delta depending only on epsilon. To disprove, use the sequential criterion: find x_n, y_n with |x_n - y_n| -> 0 but |f(x_n) - f(y_n)| >= epsilon_0. Know the two positive theorems cold: continuous on compact implies uniform, and uniformly continuous on a dense set extends uniquely.
Mishandling quantifiers and negation in epsilon-proofs: negating 'for all epsilon there exists N' incorrectly, proving the converse of an implication, or arguing by example ('take epsilon = 0.01, it works') instead of for arbitrary epsilon.
Logic is usually studied early and then abandoned; under time pressure the mechanical rules for pushing negations through alternating quantifiers decay, and a single concrete epsilon feels like it discharges the obligation.
What to do instead
Before writing, restate the goal in pure quantifier form and write its negation beside it. Begin every epsilon-proof with 'Let epsilon > 0 be arbitrary' and never substitute a number. When disproving, always produce the specific epsilon_0 and the specific index or point. Practice negating five statements per study session until it is automatic.
Using series tests without verifying their hypotheses or distinguishing absolute from conditional convergence: applying the ratio test when the limit is 1 and concluding convergence, using the alternating series test to conclude absolute convergence, or rearranging a conditionally convergent series.
The tests are learned as recipes and the hypothesis lines are skipped; the ratio test's inconclusive case is the one exam writers target, and the fact that conditional convergence forbids rearrangement is counterintuitive.
What to do instead
For each test, write the hypothesis and the conclusion (including 'inconclusive') on a card. Determine absolute convergence first with the comparison, ratio, or root test; only if that fails, try the alternating series test and label the result conditional. Never regroup or reorder terms of a conditionally convergent series without invoking a theorem that permits it.
Confusing the Riemann integral's definitional requirements: assuming every bounded function is Riemann integrable, using antiderivatives to integrate a function with a discontinuity without checking integrability, or treating an improper integral as a limit that always exists.
Computational calculus habits dominate: 'integrate' means 'find an antiderivative'. The Darboux/partition machinery and the distinction between proper and improper integrals get compressed into a formula, so discontinuities and unboundedness slip past.
What to do instead
For integrability, either exhibit partitions with U(P,f) - L(P,f) < epsilon or invoke a theorem (continuous on [a,b], or bounded with finitely many discontinuities). For improper integrals, write the limit explicitly, split at every singularity, and test convergence with comparison before computing. Keep a list of standard non-integrable or divergent examples (Dirichlet function, 1/x on (0,1], p-integrals).
Interchanging limits, derivatives, integrals, or sums without justification: differentiating a series term by term, integrating a limit, or swapping two limits, on the strength of pointwise convergence alone or with no hypothesis check at all.
In finite sums these operations are always legal, and no counterexample appears in routine computation. The theorems that license the swap (uniform convergence, dominated convergence, continuity of the limit) have hypotheses that are easy to satisfy silently in examples and hard to verify in general.
What to do instead
Never write a swap without naming its theorem in the margin: 'uniform convergence of f_n' plus 'continuous f_n' for limit-integral, 'f_n' converges uniformly and f_n' converges uniformly' for term-by-term differentiation. If a hypothesis is missing, either prove it (Weierstrass M-test is usually the tool) or find the counterexample. Memorize the standard failure: f_n(x) = x^n on [0,1] differentiated at 1.
Treating compactness as 'closed and bounded' in every setting, or proving compactness by citing Heine-Borel for a subset of an arbitrary metric space, or confusing compactness with completeness or with sequential compactness without proving equivalence in the metric case.
Heine-Borel is the only compactness theorem most students use, and R^n is the only space they picture; the exam deliberately works in C[0,1], l^2, or a function space where closed and bounded sets are not compact.
What to do instead
State which space you are in before invoking compactness. In a general metric space, prove compactness via open covers or sequential compactness and use total boundedness plus completeness as the criterion. Keep the counterexample l^2 unit ball (closed, bounded, not compact) and C[0,1] with the sup norm ready, and know that continuous image of compact is compact and continuous on compact is uniformly continuous.
Writing multivariable differentiability proofs that confuse partial derivatives with the derivative: asserting differentiability from existence of partials, or writing the inverse or implicit function theorem conclusion without checking that the relevant derivative is invertible and without identifying the neighborhood on which the conclusion holds.
In one variable, differentiability and having a derivative coincide, so the distinction vanishes; in several variables the derivative is a linear map and the counterexample (a function with partials at the origin but not differentiable there) is not part of most students' working memory.
What to do instead
Always test differentiability with the limit definition: f(a+h) - f(a) - Df(a)h divided by ||h|| must go to 0. For the inverse and implicit function theorems, state the hypotheses explicitly (C^1, derivative invertible at the point), verify invertibility by computing a determinant or exhibiting the inverse matrix, and state the conclusion locally with the neighborhood named. Know that C^1 implies differentiable and that the converse fails.