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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?

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1 Answer

  • 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.
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