Problems on Introduction to Analysis

I. Foundations

Mon, 01 Jan 0001 00:00:00 +0000

I. Foundations #

Warm-up problems on proof by induction and on the set theory of images and inverse images that underlies everything to come.

Problem 1 #

A polygon is said to be convex if it contains the line segment connecting any two of its points. A polygon can be triangulated if its vertices can be connected to each other by non-intersecting line segments in such a way that the entire polygon is divided entirely into triangles.

II. Fields & Order

Mon, 01 Jan 0001 00:00:00 +0000

II. Fields & Order #

Problems on the field and order axioms, the least upper bound property, and the structures (like $\mathbb{Q}[\sqrt{2}]$) that satisfy them.

Problem 1 #

Define the set of real numbers

$$\mathbb{Q}[\sqrt{2}] = \{\, a + b \sqrt{2} \mid a, b \in \mathbb{Q} \,\}.$$

Together with the usual addition and multiplication, $(\mathbb{Q}[\sqrt{2}], +, \cdot)$ forms a field. One could verify this fact directly by checking all the axioms of a field hold, but it turns out that is not necessary to work quite that hard. The key observation is that the set of real numbers $(\mathbb{R},+,\cdot)$ is itself a field and every element of $\mathbb{Q}[\sqrt{2}]$ is a real number.

III. Metric Spaces

Mon, 01 Jan 0001 00:00:00 +0000

III. Metric Spaces #

Problems introducing metrics and metric spaces: verifying the metric axioms, exploring the family of $p$-metrics, distances between sets, and the behavior of open balls.

Problem 1 #

As it was suggested in class, our course will focus on metrics and metric spaces. However, there are useful ways of measuring distances that do not satisfy all the properties of a metric. This exercise will introduce you to one used frequently in machine learning. We begin with a short reading about a basic problem in machine learning.

IV. Open & Closed Sets

Mon, 01 Jan 0001 00:00:00 +0000

IV. Open & Closed Sets #

Problems on the basic topology of a metric space: closed sets and their stability under unions and intersections, closures, and the interplay between closure and complementation.

Problem 1 #

Prove the following proposition.

Proposition. Let $(S,\rho)$ be a metric space. Then

(a) $S$ is a closed set;

(b) $\varnothing$ is a closed set;

(c) if $I$ is a finite set and $A_i$ is a closed set for all $i \in I$, then $\bigcup_{i \in I} A_i$ is a closed set;

V. Compactness & Connectedness

Mon, 01 Jan 0001 00:00:00 +0000

V. Compactness & Connectedness #

Problems on compact sets and their stability properties, the failure of the Heine–Borel theorem outside $\mathbb{R}$, and connectedness.

Problem 1 #

Show that the union of two compact sets is compact.

Solutions: Hint · Solution · Long solution

Problem 2 #

Suppose that $T$ is closed and $U$ is compact. Show that $T \cap U$ is compact.

VI. Sequences

Mon, 01 Jan 0001 00:00:00 +0000

VI. Sequences #

Problems on convergent sequences in metric spaces: the connection between limit points and limits of sequences, bounds and limits, and some classical estimates.

Problem 1 #

Suppose that $(S, \rho)$ is a metric space. If $T$ is a subset of $S$ and $t$ is a limit point of the set $T$, show that there is a sequence of points $\{p_n\}_{n=1}^\infty$ in $T$ such that

$$\lim p_n = t.$$

Solutions: Hint · Solution · Long solution

VII. Completeness & Fixed Points

Mon, 01 Jan 0001 00:00:00 +0000

VII. Completeness & Fixed Points #

Problems on uniform continuity and Cauchy sequences, and on contraction maps and the existence of fixed points.

Problem 1 #

Prove that every uniformly continuous function $f:S_1 \to S_2$ is continuous at every point $x_0 \in S_1$.

Solutions: Hint · Solution · Long solution

Problem 2 #

Consider a function $f:S_1 \to S_2$. If $f$ is uniformly continuous and $\{s_n\}$ is a Cauchy sequence of points in $S_1$, then the sequence $\{f(s_n)\}$ in $S_2$ is Cauchy as well. Show that this result is not true in general if $f$ is only assumed to be continuous.

VIII. Continuity

Mon, 01 Jan 0001 00:00:00 +0000

VIII. Continuity #

Problems on continuous functions between metric spaces: the basic definition, the open-set characterization, and a range of examples and constructions.

Problem 1 #

Suppose that $(S, \rho)$ is a metric space and let $f: S \to S$ be the identity function defined by $f(s)=s$ for all $s \in S$. Show that $f$ is continuous by using our original definition of continuity.

Solutions: Hint · Solution · Long solution

IX. IVT & Countability

Mon, 01 Jan 0001 00:00:00 +0000

IX. IVT & Countability #

Problems that combine continuity with density, the Intermediate Value Theorem, compactness, and cardinality arguments.

Problem 1 #

Consider a function $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(x) = 0$ for all $x \in \mathbb{Q}$. Show that if $f$ is continuous, then in fact $f(x)=0$ for all $x \in \mathbb{R}$!

Solutions: Hint · Solution · Long solution

X. Differentiation

Mon, 01 Jan 0001 00:00:00 +0000

X. Differentiation #

Problems on the derivative: its consequences for monotonicity, the existence of higher derivatives, the fact that a derivative need not be continuous, and a construction of the power rule.

Problem 1 #

Suppose that $f$ is differentiable at every point of $[a,b]$ and suppose that the derivative is never zero. Prove that $f$ is strictly monotonic on $[a,b]$. Note that $f'$ is not assumed to be continuous.

XI. Integration

Mon, 01 Jan 0001 00:00:00 +0000

XI. Integration #

Problems on upper and lower sums, criteria for Riemann integrability, and the basic order and mean-value properties of the integral.

Problem 1 #

Consider a function $f: [a,b] \to \mathbb{R}$ which may or may not be Riemann integrable. For any two partitions $P$ and $Q$ of the interval $[a,b]$, show that

$$L(f,P) \leq U(f,Q),$$

that is, any upper sum is greater than or equal to any lower sum for $f$.

XII. Uniform Convergence

Mon, 01 Jan 0001 00:00:00 +0000

XII. Uniform Convergence #

Problems on uniform convergence of sequences of functions and on which properties of the functions in a sequence are inherited by the limit.

Problem 1 #

Recall the following definition from class:

Definition. A sequence of functions $\{f_n\}$ from a set $S$ to the real numbers is said to converge uniformly to a function $f$ iff for every $\epsilon > 0$, there is an integer $N$ such that $n \geq N$ implies that