If $\{f_n\}$ is a sequence of continuous functions on $[0,1]$ such that $0 \le f_n \le 1$ and such that $f_n(x) \to 0$ as $n \to \infty$, for every $x \in [0,1]$, then \begin{equation*} \lim_{n \to \infty} \int_0^1 f_n(x)\,dx = 0. \end{equation*} Try to prove this without using any measure theory of any theorems about Lebesgue integration.

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Exercise 2.10

If ${f_{n}}$ is a sequence of continuous functions on $[0, 1]$ such that $0 \leq f_{n} \leq 1$ and such that $f_{n} (x) \to 0$ as $n \to \infty$ , for every $x \in [0, 1]$ , then

\lim_{n \to \infty} \int_{0}^{1} f_{n} (x) dx = 0 .

Try to prove this without using any measure theory of any theorems about Lebesgue integration.

Exercise 2.10

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