(a) (i) Show that a compact metric space must be complete.

(ii) If a metric space is complete and bounded, must it be compact? Give a proof or counterexample.

(b) A metric space $(X, d)$ is said to be totally bounded if for all $\epsilon>0$, there exists $N \in \mathbb{N}$ and $\left\{x_{1}, \ldots, x_{N}\right\} \subset X$ such that $X=\bigcup_{i=1}^{N} B_{\epsilon}\left(x_{i}\right) .$

(i) Show that a compact metric space is totally bounded.

(ii) Show that a complete, totally bounded metric space is compact.

[Hint: If $\left(x_{n}\right)$ is Cauchy, then there is a subsequence $\left(x_{n_{j}}\right)$ such that

$$\left.\sum_{j} d\left(x_{n_{j+1}}, x_{n_{j}}\right)<\infty .\right]$$

(iii) Consider the space $C[0,1]$ of continuous functions $f:[0,1] \rightarrow \mathbb{R}$, with the metric

$$d(f, g)=\min \left\{\int_{0}^{1}|f(t)-g(t)| d t, 1\right\} .$$

Is this space compact? Justify your answer.