1. If $x \in \R$ and there is a Cauchy sequence $(a_n)$ in $\Q$ such that $x = LIM a_n$, then show that $x = \lim a_n$.

2. For any $a \in \R$, prove that $\lim a^n / n! = 0$.

3. Let $A = \{ q \in \Q: q^2 < 3 \}$, let $x = \sup(A)$, prove that $x^2 = 3$.

4. Let $s_1 = 1$, and $s_{n+1} = \sqrt{1 + s_n}$. Assume that $s_n$ converges to $c$, show that $c=(\sqrt{5}+1)/2$.

5. Let $(a_n)$ be a bounded sequence in $\R$, and $A = \lim sup(a_n)$, show that for any $\epsilon > 0$, the set $\{ n \in \N \mid A - \epsilon \leq a_n \leq A+\epsilon \}$ is infinite.