If a function $f: A \to \mathbb R$ is bounded and continuous on $A$, does its integral on said interval converge uniformly?

Question

Let the bounded interval $A$ be a subset of the reals and let $f : A\to\mathbb R$ be a bounded and continuous function. Since the sequence $(f_n): f_n = f/n$ has the limit $\displaystyle\lim_{n\to\infty} f_n = 0$, does the integral \begin{equation} \int_A \frac{f(x)}{n}\,\mathrm d x \end{equation} converge to zero as $n\to \infty?$

Answer 1

This is simple. Let $A= [a,b].$ Suppose $|f|\le M$ on $[a,b].$ Then

$$\left|\int_a^b (f/n)\right| \le \int_a^b |(f/n)| \le \int_a^b (1/n)M = (1/n)M(b-a) \to 0.$$

Answer 2

\begin{equation} \int_A f(x)\,\mathrm d x \le \int_A \sup_x f(x) \, \mathrm d x \le \text{mes}(A) \cdot M < \infty, \end{equation} where $M := \sup_x f(x)$.