# How to prove $\frac{1}{n}$ converges to 0

by Suresh Ponnada   Last Updated August 10, 2018 11:20 AM - source

How do I prove $\frac{1}{n}$ converges to 0?

### my attempt:

for $\epsilon >0$, let $N \geq \frac{1}{\epsilon}$

if $n > N \geq \frac{1}{\epsilon}$ then $\epsilon > \frac{1}{n} = \left| \frac{1}{n} \right| = \left| \frac{1}{n} - 0\right|$

so $\frac{1}{n}\to 0$

But how do we prove it geometrically?

And how do we prove $\frac{1}{n}$ does not converge to any non zero number?

For example, if we choose $l=\frac{1}{10}$ then how do we choose a number $\epsilon$ such that $x_n=\frac{1}{n} \notin(l-\epsilon, l+\epsilon)$?

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One of the basic theorems about limits is that if a limit exists then it is unique. If your sequence converges to $0$ then it can't converge to anything else. Try to prove that theorem, it's a good exercise. And it will answer your question.
By contradiction suppose $\frac1n\to L\neq 0$ and apply the definition.