We just started learning sequences and my teacher gave us this problem that seems to be incredibly hard. I don't even know where to start.
Question: The first term of the following sequence is $1$.
$ \left\{x_{n+1}\right\}^\infty_{n=1}=\left\{\frac{x^5_n + 1}{5x_n}\right\}^\infty_{n=1} $
Show that $x_n>\frac{3}{11}$, for $n\geq1$.
Any idea? Thanks.