Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

Let $(x_i)$ be a sequence of numbers in $(0,1)$ such that $\lim_{n \to \infty} \left( \frac{1}{n} \right) \sum_{i = 1}^n x_i^k$ exists for all integers $k \geq 0$. Show that $\lim_{n \to \infty} \left( \frac{1}{n} \right) \sum_{i = 1}^n f(x_i)$ exists for all functions $f \in C[0,1]$.

I was wondering if I could get a hint?

share|cite|improve this question
  • Show that $\lim_{n\to \infty}\frac 1n\sum_{i=1}^nP(x_i)$ exists for any polynomial $P$.
  • Conclude by a well known theorem about approximation of continuous functions on a compact subset of the real line.
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.