Real Analysis Qual Prep Week 1: Preliminaries

Tags: #qualifying_exam #active_projects #qual_real_analysis

Week 1: Preliminaries


  • Continuity and uniform continuity
    • Pathological functions and sequences of functions
  • Convergence
    • The Cauchy criterion
    • Uniform convergence
    • The \(M\)-Test
  • Heine-Borel
  • Normed spaces
  • Series and sequences,
    • Convergence
    • Small tails,
    • limsup and liminf,
    • Cauchy criteria for sums and integrals
  • Variation and bounded variation

Things that don’t explicitly appear in qual problems

  • Baire category theorem,
  • Nowhere density,
  • Basic inequalities (triangle, Cauchy-Schwarz)
  • Concepts from Calculus
    • Mean value theorem
    • Taylor expansion
    • Taylor’s remainder theorem
    • Intermediate value theorem
    • Extreme value theorem
    • Rolle’s theorem
    • Riemann integrability
  • Weierstrass approximation

Background / Warmup / Review

  • Derive the reverse triangle inequality from the triangle inequality.
  • Let \(E\subseteq {\mathbf{R}}\). Define \(\sup E\) and \(\inf E\).
  • What is the Archimedean property?

Metric Spaces / Topology

  • What does it mean for a metric space to be complete?
  • Give two or more equivalently definitions for compactness in a complete metric space.
  • What is an interior point? An isolated point? A limit point?
  • What does it mean for a set to be open? Closed?
  • What is the closure of a subspace \(E\subseteq X\)?
  • What does it mean for \(E\subseteq X\) to be a dense subspace?
  • What does it mean for a family of sets to form a basis for a topology?
    • What is a basis for the standard topology on \({\mathbf{R}}^d\)?
  • Let \(X\) be a subset of \({\mathbf{R}}^d\). Prove the Heine-Borel theorem:
    • Show that \(X\) compact \(\implies X\) is closed
    • Show that \(X\) compact \(\implies X\) is bounded
    • Show that a closed subset of a compact set must be bounded.
    • Show that if \(X\) closed and bounded \(\implies X\) is compact.
  • Find an example of a metric space with a closed and bounded subspace that is not compact.
    • How can this be modified to obtain a necessary and sufficient condition?
  • Determine if the following subsets of \({\mathbf{R}}\) are opened, closed, both, or neither:
    • \({\mathbf{Q}}\)
    • \({\mathbf{Z}}\)
    • \(\left\{{1}\right\}\)
    • \(\left\{{p \in {\mathbf{Z}}^{\geq 0} {~\mathrel{\Big\vert}~}p\text{ is prime}}\right\}\)
    • \(\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\}\)
    • \(\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\} \cup\left\{{0}\right\}\)


  • Can a convergent sequence of real numbers have a subsequence converging to a different limit?
  • What does it mean for a sequence of functions to converge pointwise and to converge uniformly?
    • Give an example of a sequence that converges pointwise but not uniformly.
  • Prove that every sequence admits a monotone subsequence.
  • Prove the monotone convergence theorem for sequences.
  • Prove the Bolzano-Weierstrass Theorem.


– What does it mean for a series to converge? How can you check this?

  • What does it mean for a series to converge uniformly? What do you have to show to prove it does not converge uniformly?
  • Show that if \(\sum_{n\in {\mathbb{N}}} a_n < \infty\) converges, then `
\begin{align*}a_n \, {\overset{{n} \to\infty}\longrightarrow 0} \, \end{align*} `{=html}. - Show that convergent sequences *have small tails* in the following sense: ` \begin{align*}\sum_{n > N} a_n \, {\overset{{N} \to\infty}\longrightarrow 0} \, \end{align*} `{=html}. - Is this a necessary and sufficient condition for convergence? - State the ratio, root, integral, and alternating series tests. - Prove that the harmonic series diverges - Derive a formula for the sum of a geometric series. - State and prove the $p{\hbox{-}}$test. - What does it mean for a series to converge absolutely? - Find a sequence that converges but not absolutely.

Continuity and Discontinuity

  • What does it mean for a function to be uniformly continuous on a set?

  • Is it possible for a function \(f:{\mathbf{R}}\to {\mathbf{R}}\) to be discontinuous precisely on the rationals \({\mathbf{Q}}\)? If so, produce such a function, if not, why?

    • Can the set of discontinuities be precisely the irrationals \({\mathbf{R}}\setminus{\mathbf{Q}}\)?
  • Find a sequence of continuous functions that does not converge uniformly, but still has a pointwise limit that is continuous.


  • Find a function that is differentiable but not continuously differentiable.

  • Prove the uniform limit theorem: a uniform limit of continuous function is continuous.

  • Show that the uniform limit of bounded functions is uniformly bounded.

  • Construct sequences of functions \(\left\{{f_n}\right\}_{n\in {\mathbb{N}}}\) and \(\left\{{g_n}\right\}_{n\in {\mathbb{N}}}\) which converge uniformly on some set \(E\), and yet their product sequence \(\left\{{h_n}\right\}_{n\in {\mathbb{N}}}\) with \(h_n \coloneqq f_n g_n\) does not converge uniformly.

    • Show that if \(f_n, g_n\) are additionally bounded, then \(h_n\) does converge uniformly.
  • Find a sequence of functions such that \begin{align*}\frac{d}{d x} \lim _{n \rightarrow \infty} f_{n}(x) \neq \lim _{n \rightarrow \infty} \frac{d}{d x} f_{n}(x)\end{align*}

  • Find a uniform limit of differentiable functions that is not differentiable.

  • Prove that the Cantor set is a Borel set.

  • Show the Cantor ternary set is totally disconnected; that is show it contains no nonempty open interval.

Qual Questions

    • Note: outline omitted!

#qualifying_exam #active_projects #qual_real_analysis