Show that the sequence $\lim_{n\rightarrow\infty} \log(n)$ does not converge

Question

In order to show that the sequence $$\lim_{n\rightarrow\infty} \log(n)$$ does not converge I have to show:

there exists $N \geq 0$ s.t $$\left|a_{n} - L \right| \ > \epsilon$$ for each $N$ and for all choices of $L$.

Here is where my issue lies. I'm trying to find a relationship to apply. I asked for some help from my prof and he suggested $$ \log(n) > M \implies n > e^{M} $$

Now what comes to my mind from this is to somehow make this $M$ my $\epsilon$, but I don't see where to carry through the comparison..

Answer 1

It's enough to push the values of the sequence off to infinity, this is the purpose of the hint.

I.e. Given any positive $M$, you ought to be able to find an $N$ large enough such that for any $n>N$ $$ \log(n)>M $$ your professor told you how to find this $N$, namely $$ \log(n)>M\iff n>e^M $$ so picking $n=\lceil e^M \rceil$ will do. Since choice of $M$ was arbitrary, you are done.