Problem #PRU-61304

Problem

Method of iterations. In order to approximately solve an equation, it is allowed to write $f(x)=x$, by using the iteration method. First, some number $x_0$ is chosen, and then the sequence $\{x_n\}$ is constructed according to the rule $x_{n+1}=f(x_n)$ ($n\ge 0$). Prove that if this sequence has the limit $x\ast =\underset{n\to \mathrm{\infty }}{lim}{x}_n$, and the function $f(x)$ is continuous, then this limit is the root of the original equation: $f(x^{\ast })=x^{\ast }$.