Limits in $C[0,1]$

Question

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?

Answer 1

• 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.