Iterated function with $a_{n} = {F^{2n}(1)} $ and $b_{n} = {F^{2n+1}(1) }$. How to show that $a_n$ is increasing and $b_n$ is decreasing?

Question

Consider function $ F(x) = \sqrt{1+\frac{1}{x}}$. Let $a_{n} = {F^{2n}(1)} $ and $b_{n} = {F^{2n+1}(1) }$ . How can I show that $ a_{n} $ is increasing and $b_{n} $ is decreasing and what is their limit ?