Prove that ln(n) is not a Cauchy Sequence?

Question

Show from the definition of a Cauchy Sequence that Xn=ln(n) is not a Cauchy Series

Answer 1

You need to prove that there is some epsilon > 0 such that for any N > 0 there is some n and m with |ln(n/m)| >= epsilon

for example, is you take epsilon = 1 and n=m+1, then the inequality is satisfied after some index N

Answer 2

Assume $\log n$ is Cauchy.

For given $\epsilon >0$ there exists a $n_0$ s.t. for $m\ge n\ge n_0$

$|\log m-\log n| <\epsilon$.

Consider $m=kn$, where $k$ is a positive integer.

Then

$|\log m-\log n|= |\log (m/n)|=|\log k| <\epsilon$.

For $k$ large enough, a contradiction (Why?)