Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $|S(f:P)−L|< \epsilon$.

share|cite|improve this question
hint : what is the geometrical meaning of $y_n$ ? – Glougloubarbaki Dec 4 '11 at 23:43
I suppose you could say that $y_{n}$ is the area under the function, but I still am confused – rioneye Dec 4 '11 at 23:49
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 $1/n$. – Glougloubarbaki Dec 4 '11 at 23:56
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? – user1551 Dec 5 '11 at 0:01
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. – Srivatsan Dec 5 '11 at 0:23
up vote 1 down vote accepted

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 $\|P\|\rightarrow0$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.