given $x_1$ is positive real number, prove the sequence $x_{k+1}=\frac {1} {2} \left (x_k+ \frac {2} {x_k} \right)$ doesn't converge in Q
I started with $abs(x_n-x)<\epsilon$ and then by triangle of inequality
$\left| x_{n}-x\right| \\\left| \frac {1} {2}x_{n-1}+\frac {1} {x_{r-1}}-x\right| \leq \left| \frac {1} {2}x_{n-1}\right| +\left| \frac {1} {x_{n-1}}-x\right|$
I don't know how to proceed from here. Thanks!
If the limit exist, say it converges to $L$, then by taking $k\to \infty$ you get $L=\frac{1}{2}(L+\frac{2}{L})$, solving the equation you get $L=\sqrt{2}$. So if the limit exist, it has to be $\sqrt{2}$ which is not rational.