I am trying to prove the following:
Suppose $f$ is integrable on $[0,1]$ and define
$a_n = \frac{1}{n} \sum_{k=1}^n f \bigg(\frac{k}{n}\bigg)$
for all n. Prove $\{a_n\}_{n=1}^\infty$ converges to $\int_0^1 f dt$.
My strategy was to show the Riemann sum is equal to the limit of $\{a_n\}_{n=1}^\infty$, and I feel like I didn't exactly show that explicitly. I'm wondering if anyone can point out where I'm wrong or maybe redirect me. Here's my attempt:
Suppose $f$ is integrable on $[0,1]$ and define \begin{align*} a_n = \frac{1}{n} \sum_{i=1}^n f \bigg(\frac{i}{n}\bigg) \end{align*} Now let $\{P_n\}_{n=1}^\infty$ be a sequence of marked partitions of $[0,1]$ with $P_n = \big\{\frac{i}{n} : 0 < i \leq n\big\}$. Mark $P_n$ by letting $t_i = \frac{i}{n}$ for all $n$. Then since $f$ is integrable we have \begin{align*} S(P_n,f) = \sum_{i=1}^n f(t_i)\frac{1}{n} = \sum_{i=1}^n f \bigg(\frac{i}{n}\bigg)\frac{1}{n} = \int_0^1 f dt.\\ \end{align*} But, \begin{align*} \lim_{n \rightarrow \infty} a_n &= \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n f \bigg(\frac{i}{n}\bigg)\\ & = \lim_{n \rightarrow \infty} \sum_{i=1}^n f \bigg(\frac{i}{n}\bigg) \frac{1}{n} = S(P_n,f) \end{align*} So we have $\lim_{n \rightarrow \infty} a_n = \int_0^1 f dt$.
Thank you!