Question 20
Let $f: [0, 1] \to [0, 1]$ be a continuous function and $x_0, y_0$ be two points in $[0, 1]$. Define sequences $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ of points in $[0, 1]$ by
$$
x_{n+1} = \frac{f(x_0) + \cdots + f(x_n)}{n+1} \quad \text{and} \quad y_{n+1} = f\left(\frac{y_0 + \cdots + y_n}{n+1}\right);
$$
Clearly, the definition is correct. Prove that the sequences $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ are convergent.
$$
x_{n+1} = \frac{f(x_0) + \cdots + f(x_n)}{n+1} \quad \text{and} \quad y_{n+1} = f\left(\frac{y_0 + \cdots + y_n}{n+1}\right);
$$
Clearly, the definition is correct. Prove that the sequences $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ are convergent.