Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Define the integral $\int_{a}^{b} f(x) d x$. (You are not asked to prove existence.)

Suppose that $m, M$ are real numbers such that $m \leqslant f(x) \leqslant M$ for all $x \in[a, b]$. Stating clearly any properties of the integral that you require, show that

$$m(b-a) \leqslant \int_{a}^{b} f(x) d x \leqslant M(b-a) .$$

The function $g:[a, b] \rightarrow \mathbb{R}$ is continuous and non-negative. Show that

$$m \int_{a}^{b} g(x) d x \leqslant \int_{a}^{b} f(x) g(x) d x \leqslant M \int_{a}^{b} g(x) d x$$

Now let $f$ be continuous on $[0,1]$. By suitable choice of $g$ show that

$$\lim _{n \rightarrow \infty} \int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x=f(0),$$

and by making an appropriate change of variable, or otherwise, show that

$$\lim _{n \rightarrow \infty} \int_{0}^{1} n f(x) e^{-n x} d x=f(0) .$$