3
$\begingroup$

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 $|S(f:P)−L|< \epsilon$.

  • 2
    I edited the post and added $n \to \infty$ under the limit symbol. Check that it is ok. // Please add the relevant information **to the question**, so that they are not buried under the comments.2011-12-05

1 Answers 1

1

The problem statement says that $f$ is Riemann integrable, thus $S(f; P)\rightarrow0$ for when $\|P\|\rightarrow0$. So, all you have to do is to identify the partition $P$ (or strictly speaking, the sequence of partitions $P_n$) in your problem and show that $\|P\|\rightarrow0$.