Continued Fraction [1,1,1,...]

Solution 1:

You could prove, by induction, that $[1,1,\dots,1]=f_{n+1}/f_n$, where $f_n$ is the $n$th Fibonacci number, and then prove (using, say, the Binet formula for $f_n$) that $\lim_{n\to\infty}(f_{n+1}/f_n)=(1+\sqrt5)/2$.

Solution 2:

The only other thing you really need to show if you want to be precise is that the sequence of partial fractions given by $a_1 = [1]$, $a_2 = [1,1]$, $a_3 = [1,1,1]$, etc. does tend to a limit (it suffices to show that the sequence $\{a_n\}_{n\in\mathbb{N}}$ is bounded above by something and increasing eventually). Then your calculation shows that $\alpha$ is the unique positive solution, and hence must be equal to the infinite continued fraction (which is formally the limit of the partial fractions you get when you stop after $n$ 1's: $[1,1,1\ldots] := \lim_{n\to\infty} a_n$).