If $x_{n+1}= \frac{\sum_{i=1}^n f(x_i)}{n}$ then prove or disprove that $\{x_n\}$ is convergent.
$f:[0,1]\to[0,1]$ be a continuous function. Let $x_1\in[0,1]$ and define $x_{n+1}={\sum_{i=1}^n f(x_i)\over n}$.Prove, $\{x_n\}$ is convergent