recursive sequence $x_{n+1}=frac{1}{x_n}$

$x_{n+1}=\frac{1}{x_n}$, $x_1>1$

Find if the function converges and diverges and then prove it.

If we try and find the limit we get 1 or -1. 1 does not work because then $x_1$ and $x_n$ contradict each other. -1 works for showing its the lowest bound. I'm not sure about this proof for proving it's monotonically decreasing:

$x_n>-1$
which implies ${x_n^2}>1$ whcih implies $x_{n}>\frac{1}{x_n}$ and so $x_n>x_{n+1}$

therefore decreasing sequence