Here is the question I am struggling with:
Assume $f \in R[0, 1]$ and consider the sequence $(y_n)$, where $y_n =\frac{1}{n} \sum_{i=1}^n \; f\left(\frac{i}{n}\right) .$ Show that $\lim y_n = \int_0^1 f$.
So I can show that $y_{n} = S(f:P)$ which is the Riemann sum, but I can't figure out what I should do next. I figure I have to use the definition of a limit and somehow morph it into the definition of a Riemann integral, but I can't be sure. Any tips?
The definition of Riemann integral I am using is; there is $L \in \mathbb R$ such that for every $\epsilon > 0$ there is $\delta >0$ such that if $P$ is any tagged partition of $I$ with $\|P\|< \delta$ then $|S(f:P)−L|< \epsilon$.