Question 112
Let $f : [0, 1] \to \mathbb{R}$ be an integrable function, and let $(a_n)_{n \ge 1}$ be a bounded below sequence of real numbers, so that $\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k = a \lt \infty$. Prove that
$$
\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k f\left(\frac{k}{n}\right) = a \int_{0}^{1} f(x) dx.
$$
$$
\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k f\left(\frac{k}{n}\right) = a \int_{0}^{1} f(x) dx.
$$