Prove a sequence converges without knowing its limit

Question

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent?

I know that its limit is $6$, but the question is how to rigorously prove its convergence without knowing it.

Answer 1

HINT:

Step 1. Prove by induction that the sequence $$\left\{X_n\right\}_{n\geq1}$$ is increasing.

Step 2. Prove by induction that $$X_n\leq 6, \quad n\geq1.$$

Answer 2

Forget $X_1=2$ and look at the function $$f(x)=\sqrt{5x+6}$$ if you think about it you know that on some interval the square function is above the line $x=y$ and after some point it will be below. If at some point $x_0\in\mathbb{R}$ we have $f(x_0)>x_0$ then the sequence is increasing at that point, and if $f(x_0)<x_0$ it is deacreasing. Noticing that: $$f(x)<|x| \text{ for }x>6$$ $$f(x)>|x|\text{ for }-1<x<6$$ After the final note that $f(x)$ is increasing and as such doesn't have critical points and so will not go back and forth around the points $f(x)=x$ we get that starting at any point $X_1>-1$ will get you to have the limit $l=6$. If $-1<X_1<6$ it is increasing and if $X_1>6$ decreasing.