# $x_n^2 > 2,$ how to show that $x_{n+1}^2 > 2$?

$x_n^2 > 2,$ how to show that $x_{n+1}^2 > 2$? I have tried using induction on this but haven't been able to solve this for a while.

The sequence is defined as $x_1 = 2,$ $x_{n+1} = \frac{1}{2}(x_n + \frac{2}{x_n}).$ All I got by induction was that 2 > 1.5, which is not sufficient (I just squared and expanded the terms). How can I solve this problem?

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HINT: Use the AM-GM inequality. Alternatively, you can do: \begin{align*} x_{n+1}^2 = \frac{1}{4} \left( x_n^2 + \frac{4}{x_n^2} + 4 \right) = \frac{1}{4} \left[ \left( x_n - \frac{2}{x_n} \right)^2 + 4 + 4 \right] \gt \cdots \end{align*} (Complete the chain of (in)equalities.)