Discussing the convergence of a sequence

Question

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 = x + k$.

Answer 1

The function $f(x)=\sqrt{k+x}, x\ge 0$ is an increasing function. Therefore the sequence $(x_n)$ defined as $x_{n+1}=f(x_n)$ is convergent.

There’s no discussion sobre $x_1$ or anything else. But the monotony of that sequence depend on $x_1$.

By the way, if $x_1\leq \alpha$ $(x_n)$ becomes an increasing sequence. If not it becomes a decreasing sequence. In both cases the limit is $\alpha$.