Let $k>1$ and define a sequence $\left\{a_{n}\right\}$ by $a_{1}=1$ and $$a_{n+1}=\frac{k\left(1+a_{n}\right) }{\left(k+a_{n}\right)}$$ (a) Show that $\left\{a_{n}\right\}$ converges.
(b) Find $\lim a_{n}$
I have no problem finding the limit by taking the limit of both sides and then solving for $L=\pm\sqrt{k}$. I am not sure how to go about showing that it does in fact converge and to which value the limit actually is?
I have tried showing it is bounded monotonic sequence but have not been able to. Also I wonder if trying to prove it is Cauchy would be useful?
This is for an intro Real Analysis course so we are using basic techniques.