Week 3 | Nicholas Hu

Precompactness

Exercise 3.1

Let $(X, d)$ be a metric space and $Y \subseteq X$ . Show that $Y$ is precompact (that is, $\overset{―}{Y}$ 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 $d_{Y} (f (x_{n}), f (x_{n}^{'})) \to 0$ .