Proof and Real Analysis — Practice Problems

Practice problems for the Proof and Real Analysis lecture
Author

William Duan

These practice questions accompany the Proof and Real Analysis lecture. Each asks you to prove a result stated in the lecture. A few of the harder proofs are marked (Challenging) — attempt the others first.

Question 1

Prove De Morgan’s laws for sets: for \(A, B \subseteq X\), \[(A \cup B)^c = A^c \cap B^c \qquad\text{and}\qquad (A \cap B)^c = A^c \cup B^c.\]

Question 2

Prove the following stability properties of open and closed sets in \({\mathbb{R}}^n\).

  1. The union of an arbitrary family of open sets is open.
  2. The intersection of finitely many open sets is open.
  3. Using De Morgan’s laws, deduce that an arbitrary intersection of closed sets is closed, and a finite union of closed sets is closed.

Question 3

Let \(A\subseteq{\mathbb{R}}\) be non-empty and let \(s\in{\mathbb{R}}\) be an upper bound of \(A\). Prove that \(s = \sup A\) if and only if \[\forall\, \varepsilon>0,\ \exists\, a\in A \text{ such that } a > s-\varepsilon.\]

Question 4

Using the definition compact \(=\) closed and bounded (in \({\mathbb{R}}^n\)), justify each claim.

  1. \([0,1]^n\) is compact.
  2. \((0,1)\) is not compact.
  3. \({\mathbb{R}}\) is not compact.
  4. The budget set \(\{x\in{\mathbb{R}}^n : x\ge 0,\ p\cdot x \le w\}\) is compact, where \(p\gg 0\) (all prices strictly positive) and \(w>0\).

Question 5

Prove that if a sequence \((x_k)\) in \({\mathbb{R}}^n\) converges, then its limit is unique.

Question 6

Let \(x_k\to x\) and \(y_k\to y\) in \({\mathbb{R}}\). Prove:

  1. (Algebraic Limit Theorem) \(x_k+y_k\to x+y\); \(\ c\,x_k\to cx\); \(\ x_k y_k\to xy\); and, if \(y\ne0\) and \(y_k\ne0\), \(\ x_k/y_k\to x/y\).
  2. (Order Limit Theorem) if \(x_k\le y_k\) for all \(k\), then \(x\le y\).
  3. (Squeeze Theorem) if \(x_k\le z_k\le y_k\) for all \(k\) and \(x_k,y_k\to L\), then \(z_k\to L\).

The quotient limit in (a) is the most delicate part.

Question 7

  1. Prove the Monotone Convergence Theorem: every bounded, monotone sequence in \({\mathbb{R}}\) converges. (Prove the non-decreasing case; the non-increasing case is symmetric.)
  2. Show directly from the \(\varepsilon\)-\(K\) definition — without invoking the MCT — that \(x_k = 1-\tfrac1k\) converges to \(1\).

Question 8

  1. Prove the sequential characterisation of closed sets: \(F\subseteq{\mathbb{R}}^n\) is closed if and only if every convergent sequence \((x_k)\subseteq F\) has its limit in \(F\).
  2. Use it to show \([0,1]\) is closed and \((0,1)\) is not.

Question 9

  1. Prove that if \(x_k\to L\), then every subsequence \((x_{k_j})\) converges to \(L\).
  2. (Challenging.) Prove the Bolzano–Weierstrass theorem: every bounded sequence in \({\mathbb{R}}\) has a convergent subsequence. Hint: first show every real sequence has a monotone subsequence, then apply the Monotone Convergence Theorem.

Question 10

Let \(f:X\to Y\) and \(A,B\subseteq Y\). Prove the preimage identities: \[ f^{-1}(A\cup B)=f^{-1}(A)\cup f^{-1}(B),\quad f^{-1}(A\cap B)=f^{-1}(A)\cap f^{-1}(B),\quad f^{-1}(A^c)=\big(f^{-1}(A)\big)^c. \]

Question 11

Prove that the \(\varepsilon\)-\(\delta\) and sequential definitions of continuity at a point are equivalent: for \(f:X\to{\mathbb{R}}^m\) and \(x_0\in X\), \[ \big(\forall\varepsilon>0\ \exists\delta>0:\ \|x-x_0\|<\delta\Rightarrow\|f(x)-f(x_0)\|<\varepsilon\big) \iff \big(x_k\to x_0\Rightarrow f(x_k)\to f(x_0)\big). \]

Question 12

Let \(f,g:X\to{\mathbb{R}}\) be continuous at \(x_0\), and let \(c\in{\mathbb{R}}\).

  1. Using the sequential characterisation of continuity, prove that \(f+g\), \(cf\), and \(fg\) are continuous at \(x_0\), and that \(f/g\) is continuous at \(x_0\) provided \(g(x_0)\ne0\).
  2. Prove that a composition of continuous functions is continuous: if \(f:X\to Y\) is continuous at \(x_0\) and \(g:Y\to Z\) is continuous at \(f(x_0)\), then \(g\circ f\) is continuous at \(x_0\).

Question 13

(Challenging.)

  1. Prove the Extreme Value Theorem: if \(K\subseteq{\mathbb{R}}^n\) is compact and \(f:K\to{\mathbb{R}}\) is continuous, then \(f\) attains a maximum and a minimum on \(K\).
  2. Prove the Intermediate Value Theorem: if \(f:[a,b]\to{\mathbb{R}}\) is continuous and \(y\) lies between \(f(a)\) and \(f(b)\), then \(f(c)=y\) for some \(c\in[a,b]\).