Prove that the sequence$(x_n)=\frac{1}{2}(x_n+\frac{a}{x_n})$ where $x_1=1$ and $a>1$ is convergent

Question

So I have a sequence $(x_n)=\frac{1}{2}(x_n+\frac{a}{x_n})$ where $a>1$ and $x_1=1$.

I have to prove it is convergent.I tried with induction to prove that

$a\le x_n^2$ because from that I can easily prove both monotonicity and that it is a bounded sequence.

I check for $x_2$, it works.

Assume it will be true for $n=k$

$a \le x_k^2$

Then prove it for $n = k+1$

$a \le x_k^2$

$a \le (\frac{1}{2}(x_k+\frac{a}{x_k}))^2$

$a \le \frac{1}{4}(x_k^2+2a+\frac{a^2}{x_k^2})$

I am not sure how to prove from this step. I should get somehow $2a \le x_k^2 + \frac{a^2}{x_k^2}$ but I am unsure how.

Thank you for your help in advance.

Answer 1

I solved it with $G_M \le A_M$ inequality.

Take $\sqrt {x_k^2 \frac {a^2}{x_k^2}} \le \frac{x_k^2 + \frac{a^2}{x_k^2}}{2}$

And it will simplify to the needed inequality.