Problem: Suppose $f\in R(x)$ on $[0,1]$. Define $$a_n=\frac{1}{n}\sum\limits_{k=1}^{n}{f(\frac{k}{n})}$$
For all $n$. Prove that $\{a_n\}_{n=1}^{\infty}$ converges to $\int_0^1f\text{ } dt$.
I'm not sure if I'm supposed to formulate a partition and show that this function is bounded. Since the function $f\in R(x)$, it is automatically continuous, but I don't know if that's helpful. I'm not sure if I have to do an $\epsilon$, $\delta$ proof for this because I'm under the impression that those are essential for proofs if the function is bounded. We have no guarantee this function is bounded so I guess that would leave 2 cases. Perhaps it is supposed to be more simple?