Let $\left(s_{n}\right)_{n=1}^{\infty}$ be a sequence of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ and let $s: \mathbb{R} \rightarrow \mathbb{R}$ be another continuous function. What does it mean to say that $s_{n} \rightarrow s$ uniformly? Give examples (without proof) of a sequence $\left(s_{n}\right)$ of nonzero functions which converges to 0 uniformly, and of a sequence which converges to 0 pointwise but not uniformly. Show that if $s_{n} \rightarrow s$ uniformly then

$$\int_{-1}^{1} s_{n}(x) d x \rightarrow \int_{-1}^{1} s(x) d x$$

Give an example of a continuous function $s: \mathbb{R} \rightarrow \mathbb{R}$ with $s(x) \geqslant 0$ for all $x, s(x) \rightarrow 0$ as $|x| \rightarrow \infty$ but for which $\int_{-\infty}^{\infty} s(x) d x$ does not converge. For each positive integer $n$ define $s_{n}(x)$ to be equal to $s(x)$ if $|x| \leqslant n$, and to be $s(n) \min \left(1,|| x|-n|^{-2}\right)$ for $|x|>n$. Show that the functions $s_{n}$ are continuous, tend uniformly to $s$, and furthermore that $\int_{-\infty}^{\infty} s_{n}(x) d x$ exists and is finite for all $n$.