Explain what it means for a sequence of functions $\left(f_{n}\right)$ to converge uniformly to a function $f$ on an interval. If $\left(f_{n}\right)$ is a sequence of continuous functions converging uniformly to $f$ on a finite interval $[a, b]$, show that

$$\int_{a}^{b} f_{n}(x) d x \longrightarrow \int_{a}^{b} f(x) d x \quad \text { as } n \rightarrow \infty$$

Let $f_{n}(x)=x \exp (-x / n) / n^{2}, x \geqslant 0$. Does $f_{n} \rightarrow 0$ uniformly on $[0, \infty) ?$ Does $\int_{0}^{\infty} f_{n}(x) d x \rightarrow 0$ ? Justify your answers.