I was thinking about the following problem/proof:
In (R, $d_{2}$) let there be a convergent sequence $x_{n}$ $\Rightarrow$ $x$. Prove the following result:
$1/x_{n}$ $\rightarrow$ $1/x$ if $x\neq0$.
I am thinking about this:
If $\epsilon>0$ and we want to show that $\exists$ $N \in \mathbb{N}$ such as |1/$x_{n}-1/x$|<$\epsilon$ for any $n>\mathbb{N}$.
Knowing that $x_{n}\rightarrow x$, then $\exists$ N such as |$x_{n}-x$|< ???
How can I derive/find the threshold stated in ?. I am stuck in this part and would like to write the complete proof.
Thank you