Let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function.

(a) Let $m=\min _{x \in[a, b]} f(x)$ and $M=\max _{x \in[a, b]} f(x)$. If $g:[a, b] \rightarrow \mathbb{R}$ is a positive continuous function, prove that

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

directly from the definition of the Riemann integral.

(b) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. Show that

$$\int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x \rightarrow f(0)$$

as $n \rightarrow \infty$, and deduce that

$$\int_{0}^{1} n f(x) e^{-n x} d x \rightarrow f(0)$$

as $n \rightarrow \infty$