Does the recurring succession $X_{n+1} = 1+\frac{1}{X_n}$ converge? I'm working on some calculus exercises and came across this one. It is familiar yet not the same as the $\mathcal e$ one. The question states:
Given $X_0 = 1$ and $X_{n+1} = 1+\frac{1}{X_n}$, for $n \geq 0$.
Show that it converges and find its limit. 

So I tried to approach this by looking at its behave. Does $\frac{1}{X_n}$ converge to $0$ ? But it's not trivial answer. I also tried to look at $X_{n+1} $ expressed as a function of $X_0$ but was hard to see if I could simplify it in one-line formula. Any ideas? 

It does converge. Here is an sketch of how one can proceed. From definition, it is easy to check that
$$1\leq X_n\leq2$$
The right-hand side for example follows by induction from $1\leq X_n$ for then $X_{n+1}=1+\frac{1}{X_n}\geq2$.
It is easy to check that $X_{2n}\leq X_{2(n+1)}$ and $X_{2(n+1)-1}\leq X_{2n-1}$.
From this it you can check that $X_n$ converges an the limit must satisfy
$X=1+\frac{1}{X}$, that is, $X^2-X-1=0$. The solution that we look for is $X=\frac{1+\sqrt{5}}{2}$. 
