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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.

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    You can show by induction that $\dfrac{2}{5}\leq x_n\leq \dfrac{3}{5}$ for all $n\geq 2$.2012-11-15

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