Precompactness
Exercise 3.1
Let $(X, d)$ be a metric space and $Y \subseteq X$. Show that $Y$ is precompact (that is, $\overline{Y}$ is compact) if and only if every sequence in $Y$ has a subsequence that converges to a point in $X$.
Uniform continuity
Exercise 3.2
Let $f : X \to Y$ be a function between metric spaces $(X, d_X)$ and $(Y, d_Y)$. Show that $f$ is uniformly continuous if and only if for all sequences $\{x_n\}_{n=1}^\infty, \{x_n’\}_{n=1}^\infty \subseteq X$ with $d_X(x_n, x_n) \to 0$, we have $d_Y(f(x_n), f(x_n’)) \to 0$.