How to prove given sequence converges to 0?

Question

I am given a sequence {$a_{n}$} of nonzero numbers converges to infinity. How can I use this to prove that the sequence {$\frac{1}{a_{n}}$} converges to 0?

I can intuitively see why $\frac{1}{\infty}$ would converge to 0 but have no idea how to go about proving this. The working definition I have for convergence is $|x_{n} - L| < \varepsilon$ if for each $\varepsilon > 0$ $\exists N\in N$ such that the inequality is true $\forall n \geq N$.

Any hints/help would be much appreciated! Thank you.

Answer 1

There exists $n_0$ such that $|a_n| >\frac 1 {\epsilon}$ for $n \geq n_0$. Hence $|\frac 1 {a_n}| <\epsilon $ for $n \geq n_0$.