I.

1. $A, B$ and $C$ are any three sets. $|A| = 10$, $|B| = 20$, $|A \cup B| = 25$, $|A \cap C| = 5$, $|B \setminus C| = 10$ and $|C \setminus B| = 15$. Find
   • $|A \cap B|$
   • $|B \setminus A|$
   • $|B \cup C|$
   • $|A \cup C|$
   • $|C|$
2. The Fibonacci sequence is the sequence of numbers $0,1,1,2,3,5,8,13,\cdots$. Formally, it is defined as $F_0 = 0$, $F_1 = 1$, $F_n = F_{n-2} + F_{n-1}$. Show that $F_n^2 + F^2_{n+1} = F_{2n+1}$
3. Show that the relation $R$ on $\naturals$, defined as $x R y \Leftrightarrow y = 2^nx$ for some $n \geq 0$, is an order relation.
4. For all $x \in \reals$, show that there exists a $n \in \integers$ such that $n-1 \leq x \leq n$.
5. Suppose $z = a + ib$, $w = u + iv$, where $i = \sqrt{-1}$. Let \[ a = \left(\frac{|w| + u}{2}\right)^{\frac{1}{2}},\,\,\,\,\,\,\, b = \left(\frac{|w| - u}{2}\right)^{\frac{1}{2}} \]

   Show that $z^2 = w$ if $v \geq 0$ and that $\left(\bar{z}\right)^2 = w$ if $v \leq 0$. Conclude that every complex number, with one exception, has two square roots.

II.

1. Let $A$ be a countable set. Show that $A^n$ is countable (for any positive integer $n$).
2. Are the following metrics? Prove your answers.
   • $d(x,y) = (x - y)^2$
   • $d(x,y) = \sqrt{|x - y|}$
   • $d(x,y) = |x^2 - y^2|$
   • $d(x,y) = |x - 2y|$
   • $d(x,y) = \frac{|x - y|}{1 + |x-y|}$
3. We define the $\epsilon$ neighborhood of a point $x$ in any metric space $\mathcal{M}$ as $N_\epsilon(x) = \{y | y \in \mathcal{M}\,and\, d(x,y) \lt \epsilon\}$. Show that every neighborhood is an open set.
4. Let $A$ and $B$ be separated subsets of some $\reals^k$. Let $\mathbf{a} \in A$ and $\mathbf{b} \in B$. Define \[ p(t) = (1-t) \mathbf{a} + t \mathbf{b} \] for $t \in \reals$. Let $A_0 = p^{-1}(A)$, i.e. $A_0 = \{t | p(t) \in A\}$. Similarly, let $B_0 = p^{-1}(B)$.
   • Show that $A_0$ and $B_0$ are separated subsets of $\reals$.
   • Show that there exists $t_0 \in (0,1)$ such that $p(t_0) \notin A \cup B$.

III.

1. Show that for $\lim_{n \rightarrow \infty} \sqrt[n]{n} - 1 = 0$. Hint: you may need the binomial theorem.
2. Show that if $\sum_n a_n$ converges, then $\lim_{n \rightarrow \infty} a_n = 0$.
3. Verify if the following sequences satisfy the Cauchy criterion for convergence.
   • $x_1 = 1$, $x_{n+1} = 1 + \frac{1}{x_n} \,\,\forall\, n\geq 1$.
   • $x_1 = 1$, $x_{n+1} = \frac{1}{2 + x_n^2} \,\,\forall\, n\geq 1$.
   • $x_1 = 1$, $x_{n+1} = \frac{1}{6}(8 + x_n^2) \,\,\forall\, n\geq 1$.
4. let $\{p_n\}$ is a Cauchy sequence in a metric space $X$, and some subsequence $\{p_{n_k}\}$ converges to a point $p \in X$. Then show that the entire sequence $\{p_n\}$ converges to $p$.

IV.

1. Let $\mathcal{X}$ and $\mathcal{Y}$ be two metric spaces. Let function $f$ be a mapping from $E \in \mathcal{X}$ to $\mathcal{Y}$. Let $p$ be a limit point of $E$, and let $\lim_{x \rightarrow p} f(x) = q$. Show that $q$ is unique (i.e. there is no other point $q'$ such that $\lim_{x \rightarrow p} f(x) = q'$).
2. Show that $f(x) = log(1 + exp(x))$ is a continuous function from $\reals$ to $\reals$.
3. Every rational number $x$ can be written as $x = m/n$, where $n \gt 0$, and $GCD(m,n) = 1$. For $x = 0$ we take $n = 1$. Let $f$ be a map from $\reals$ to $\reals$ defined by \[ f(x)= \begin{cases} 0 &\mbox{$x$ is irrational}\\ \frac{1}{n} &\,\,x = \frac{m}{n}\,\,m,n \in \integers,\,n \gt 0 \end{cases} \] Show that $f$ is continuous at every irrational point.
4. Let $E$ be a non-empty subset of a metric space $\mathcal{X}$. Define the distance from $x\in \mathcal{X}$ to $E$ as \[ \rho_E(x) = \inf_{z \in E} d(x,z) \]
   • Show that $\rho_E(x) = 0$ if and only if $x \in \bar{E}$ (the closure of $E$).
   • Show that $\rho_E(x) = 0$ is a uniformly continuous function on $\mathcal{X}$ by showing that \[ |\rho_E(x) - \rho_E(y)| \leq d(x,y) \]

V.

1. Suppose $f$ and $g$ are two functions defined on $[a,b]$ (in $\reals$), and are differentiable at a point $x \in [a,b]$.
   • Show that $fg$ is differentiable at $x$.
   • Show that $fg'(x) = f'(x)g(x) + f(x)g'(x)$
   • Show that $f/g$ is differentiable at $x$ (assume $g(x) \neq 0$).
   • Show that \[ \left(\frac{f}{g}\right)'(x) = \frac{g(x)f'(x) - g'(x)f(x)}{g^2(x)} \]
2. Suppose $f'(x) \gt 0$ in $(a,b)$.
   • Prove that $f$ is strictly increasing in $(a,b)$.
   • Let $g$ be the inverse function of $f$. Show that $g$ is differentiable, and that \[ g'(f(x)) = \frac{1}{f'(x)} \,\,(a\lt x\lt b) \]
3. Suppose $f'(x)$ and $g'(x)$ exist, $g'(x) \neq 0$, and $f(x) = g(x) = 0$. Show that \[ \lim_{t \rightarrow x} \frac{f(t)}{g(t)} = \frac{f'(x)}{g'(x)} \]
4. Suppose $a \in \reals$, $f$ is a twice-differentiable real function on $(a, \infty)$, and $M_0, M_1, M_2$ are the least upper bounds of $|f(x)|$, $|f'(x)|$ and $|f"(x)|$ respectively on $(a, \infty)$. Show that \[ M^2_1 \leq 4M_0 M_2 \]