If $x$ and $A$ are positive and $x_1 = \frac{1}{2}(x + A/x)$,$x_2 = \frac{1}{2}(x_1 + A/x_1)$, and so forth, prove that the sequence $x_n$ is convergent and determine its limit.
Use your result to find an approximation to $\sqrt2$
My attempt:
To prove the convergence: I showed that since $x_n = \frac{1}{2}(x_{n-1} + A/x_{n-1}) \geq \sqrt{A}$ using the geometric-arithmetic inequality that $x_n$ is bounded below.
Then, I did $x_{n+1}-x_{n}=\frac{1}{2}(A/x_{n}-x_n) \leq 0 $ using the fact that $x_n \geq \sqrt{A} \space for \space all\space n$
Which implies that $x_n$ is monotone decreasing. So we can conclude that it is convergent.
But how can I get an approximation of $\sqrt2$ using my result?
Any help would be appreciated. Thank you!
