# Limit of a sequence does not goes to zero

## Homework Statement

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.

## 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

Dick
Homework Helper

## Homework Statement

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.

## 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

That looks just fine to me.

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