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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 $|S(f:P)−L|< \epsilon$.

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  • $\begingroup$ hint : what is the geometrical meaning of $y_n$ ? $\endgroup$ – Glougloubarbaki Dec 4 '11 at 23:43
  • $\begingroup$ I suppose you could say that $y_{n}$ is the area under the function, but I still am confused $\endgroup$ – rioneye Dec 4 '11 at 23:49
  • $\begingroup$ well not exactly, the area under the function is its integral. $y_n$ is what you obtain with the rectangular method of approximation, with a step of $1/n$. $\endgroup$ – Glougloubarbaki Dec 4 '11 at 23:56
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    $\begingroup$ There are several (equivalent) textbook definitions of a Riemann integral, but different definitions lead to different steps in solving the problem, and hence different answers to your question. How is Riemann integral defined in your class? $\endgroup$ – user1551 Dec 5 '11 at 0:01
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    $\begingroup$ 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. $\endgroup$ – Srivatsan Dec 5 '11 at 0:23
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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 $\|P\|\rightarrow0$.

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