# Week 1: Preliminaries

## Topics

• 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:
• $${\mathbf{Q}}$$
• $${\mathbf{Z}}$$
• $$\left\{{1}\right\}$$
• $$\left\{{p \in {\mathbf{Z}}^{\geq 0} {~\mathrel{\Big\vert}~}p\text{ is prime}}\right\}$$
• $$\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\}$$
• $$\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\} \cup\left\{{0}\right\}$$

### Sequences

• 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.

### Series

– 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.

## Exercises

• 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!