Suppose that $\displaystyle \{ X_{n}\}_{n=1}^{\infty}$ is a sequence of real numbers that converges to $\displaystyle x_{0}$ and that all $\displaystyle x_{n}$ and $\displaystyle x_{0}$ are non-zero.

(i) Prove that there is a positive number $\displaystyle B$ that $\displaystyle |x_{n}|\geq B$ for all n.

(ii)Using part (i), prove that $\displaystyle \{\frac{1}{x_{n}}\}$ converges to $\displaystyle \{\frac{1}{x_{0}}\}$

OK, so here is how I did part (i):

Let $\displaystyle |x_{s}|$ be the smallest absolute value in the given sequence, i.e. $\displaystyle |x_{s}|=min(|x_{n}|) \forall n$

For $\displaystyle \epsilon\geq 0$,

In $\displaystyle N_{\epsilon}(|x_{s}|)$ there exists a number y such that $\displaystyle y\leq x_{s}\to y\leq |x_{n}|$ for all n.

(ii) [I have no idea how to do this one -- or even that the first part is right. So any help will be greatly appreciated. Thanks in advance]