Check proof that if $a_{n+1} = \frac{1}{k}(a_n+\frac{k}{a_n})$ with $k>1$ and $a_1>0$, then $a_n\to\sqrt {\frac{k}{k-1}}$
Convergence of recursive sequence $a_{n+1} =\frac{ 1}{k} \left(a_{n} + \frac{k}{a_{n}}\right)$