Uniform convergence
Exercise 4.1
Let $\{f_n\}_{n=1}^\infty$ and $\{g_n\}_{n=1}^\infty$ be sequences of real-valued functions on a metric space $X$ that converge uniformly on $X$. Must $\{f_n + g_n\}_{n=1}^\infty$ converge uniformly on $X$? What about $\{f_n g_n\}_{n=1}^\infty$?
Uniform continuity
Exercise 4.2
For each $k \in \mathbb{N}$, let $f_k : \ell^2 \to \mathbb{R}$ be given by $f_k(\{x_n\}_{n=1}^\infty) = x_k$. Is $f_k$ continuous for a given $k$? Is it uniformly continuous?
(Here $\ell^2$ denotes the set of real-valued square-summable sequences endowed with the metric $d_{\ell^2}(\{x_n\}_{n=1}^\infty, \{y_n\}_{n=1}^\infty) = \left(\sum_{n=1}^\infty (x_n - y_n)^2\right)^{1/2}$.)