Exercise 5.1: Failure of Total Boundedness
For all $j \in \N$, let $f_j\colon[0,1] \to [0, 1]$ pass through the coordinates $(0,1)$, $(2^{-j}, 0)$, and $(1, 0)$. Then $\norm{f_j - f_{j+1}}_\infty \ge \tfrac12$ for all $j \in \N$, so, noting that $f_j \ge f_{j + 1}$, we have $\norm{f_j - f_{k}}_\infty \ge \tfrac12$ for all $j, k \in \N$. Since $(f_j)_{j \in \N} \sub \mathcal{C}([0, 1], b)$, it follows that $\mathcal{C}([0, 1], b)$ is not totally bounded, for otherwise the space could be covered with a finite number of $\tfrac18$-balls, meaning that $\norm{f_j - f_{k}} \le \tfrac14$ for some $j, k \in \N$.