Let $x,y\in\mathbb{Q}$ with $x>0,y>0$ and $f:\mathbb{N}\rightarrow\mathbb{Q}$ where $f(0)=1$ and $f(n+1)=\frac{1}{2}\left(f(n)+\frac{4}{f(n)}\right)$. I want to show that f converges in $\mathbb{Q}$.
For this, I used that $f(n)>0\; \forall n\in\mathbb{N}$, which implies $f(n)\geq 2$ for all $n\neq 0$ and further $f(n)^2\geq 4$ for all $n\neq 0$. Using this I showed that f is monotonically decreasing. So in $\mathbb{R}$ the limit exists (because the sequence is bounded). By using that $\lim_{n\mapsto \infty} f(n+1)=\lim_{n\mapsto \infty} f(n)$ i got an equation which I then solved for the limit.
Long story short: How can I prove that $f$ converges in $\mathbb{Q}$ (and what the limit was) without using anything that has to do with $\mathbb{R}$?
My idea for proving that $f$ converges is, to show that f is a cauchy-sequence (in $\mathbb{Q})$. Can anyone assist in finding a way to do so? I am stuck...
Thanks!