Exercise 10, § 2

a) Let $X$ be a compact space, let $(f_n)$ be a sequence of continuous mappings $X \to X$, which converges pointwise but not uniformly in $X$ to a continuous function (§ 1, no. 6, Remark 2). Show that the set of mappings $f_n$ is relatively compact in $C_s(X; X)$ but not in

b) Let $I$ be the interval $[-\mathbf{1}, +\mathbf{1}]$ of $\mathbf{R}$ and for each integer $n > 0$ let $u(x_n) = \sin \sqrt{x + 4n^2 \pi^2}$ for all $x \geq 0$. Show that the set $H$ of functions $u_n$ is an equicontinuous subset of $C(\mathbf{R}_+; I)$ and is relatively compact in $C_c(\mathbf{R}_+; I)$ but not in $C_u(\mathbf{R}_+; I)$. (Note that the sequence $(u_n)$ converges pointwise to 0.)