Prove that if $f(x)$ is integrable between $0$ and $1$, then \begin{equation} \lim_{n\to\infty}\,\frac{1}{n}\sum_{k=1}^n\,f\left(\frac{k}{n}\right)=\int_0^1f(x)\,dx. \end{equation}
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Lesson Parent:
Problem:
Prove that if $f(x)$ is integrable between $0$ and $1$, then
\begin{equation}
\lim_{n\to\infty}\,\frac{1}{n}\sum_{k=1}^n\,f\left(\frac{k}{n}\right)=\int_0^1f(x)\,dx.
\end{equation}
