Uniform convergence
Exercise 6.1 [Homework 3, Problem 2(c)]
Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. Show that $B(X, Y)$ is complete if $Y$ is.
(Here $B(X, Y)$ denotes the set of bounded functions from $X$ to $Y$ endowed with the metric $d_{B(X, Y)}(f, g) = \sup_{x\in X} d_Y(f(x), g(x))$.)
Exercise 6.2 [Homework 3, Problem 5]
Let $\{f_n\}_{n=1}^\infty$ be a sequence of functions from a metric space $(X, d_X)$ to a metric space $(Y, d_Y)$ and $f: X \to Y$ be a function. Prove that $f_n \to f$ uniformly if and only if $d_Y(f_n(x_n), f(x_n)) \to 0$ for every sequence $\{x_n\}_{n=1}^\infty \subseteq X$.