Proving Heron's sequence

Question

I'm trying to work out the following theorem.

The limit of the Heron's sequence is $\sqrt{2}$, with $x_{0}=1$, and $x_{n}= \frac{x_{n-1}}{2}+\frac{1}{x_{n-1}}$.

I'm guessing I'll need the axiom of real numbers and thereby I need to show that the sequence is non-increasing, which is easily done, but also that there exists a lower bound. I can use $L=0$ for this, since the elements will never be negative. Thus I now know there must exist a largest lower bound.

But how to go from there? That means there exists $L \in \mathbb{R}$ such that

$$|x_{n}-L| < \epsilon$$ for all $\epsilon>0$ and $n \geq N$ for some $N \in \mathbb{N}$.

Any suggestions on how to continue working on this proof?

Answer 1

Hint: Use the AM-GM inequality, to show that $$\frac{x_{n-1}}{2}+\frac{1}{x_{n-1}}\geq 2\sqrt{\frac{x_{n-1}}{2}\cdot \frac{1}{x_{n-1}}}=\frac{2}{\sqrt{2}}=\sqrt{2}$$ The $x_i$ are assumed to be positive.