I want to show that $x_1=2,x_{n+1}=\frac{x_n^2+2}{2x_n}$ is a convergent sequence and thus want to bound it below by $\sqrt{2}$. I tried to use induction but my problem is that if $x_n>\sqrt{2}$ then the denominator is greater than 4 but the denominator is less than $\frac{1}{2\sqrt{2}}$ so they compete against each other. How can solve this issue?
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