# Limit of a sequence does not goes to zero

1. Nov 1, 2013

### Seydlitz

1. The problem statement, all variables and given/known data
Could you guys please verify this proof of mine? I want to show that a limit with particular property does not go to 0. It is part of the proof that when a sequence have an ever increasing term then the limit of the sequence is not 0.

3. The attempt at a solution

The $\lim_{n \to \infty} a_n \neq 0$ if $|a_n|<|a_{n+1}|$

Suppose $\lim_{n \to \infty} a_n = 0$, then there exist $n>0$, such that $|a_N| < \epsilon$ when $N>n$. Taking an arbitrary $a_n$ as $\epsilon$, we can get $|a_N| < |a_n|.$ Because it also true that $N=n+1>n$, we get $|a_n|>|a_{n+1}|$. A contradiction.

Hence it is not possible for the limit to be 0.

Thank You

2. Nov 1, 2013

### Dick

That looks just fine to me.

3. Nov 1, 2013

### Seydlitz

Ok thanks for your verification Dick! I'm quite happy to discover this by myself.