I'm sorry if this is a duplicate. I have no idea on what kind of "name" i should give to this, and therefore i have no idea on how to search on the internet for help on understanding it. If it happens that this is a duplicate, i would be grateful if you could link me to where there are any solutions for this.
I need to prove for an exercise on my analysis book that the following sequence $$ {\cfrac{1}{1+\cfrac{1}{5}}},\quad {\cfrac{1}{1+\cfrac{1}{5+\cfrac{1}{1+\cfrac{1}{5}}}}},\dotsc $$
is monotone and converges to ${\frac{-5+\sqrt{45}}{2}}$
I imagine that once i get on how to determine it's limit, it will be easy to prove that it is in fact monotone. I have no idea on how to approach it though. Any tips?