Sequence $\ln(n)/n$ to $0$ I'm taking a real analysis course for my second year and I'm still new when it comes to convergence of a sequence using the M-epsilon definition.
I've simplified my sequence up to
$$\frac{\ln(n)}{n}<\epsilon,$$
but where do I go from here? How do I find an upper bound for $\ln(n)/n$, I have tried using $\epsilon/n$ and $(n-1)/n$ ,but it doesn't satisfy the inequality for a large $n$.

To find the maximum of a sequence consider it as a function, namely
$$f(x) = \frac{\ln x}{ x}.$$
Assuming $x>0$ we find the extremum of this function
$$f'(x) = \frac{1-\ln x}{x^2}=0 \Rightarrow \ln x =1 \Rightarrow x=e.$$
Then
$$\sup_{x \in [1,\infty)} \frac{\ln x}{x} \leq \frac{1}{e} \Rightarrow | \frac{\ln n}{n}| \leq \frac{1}{e} \quad \forall n > 0 .$$
The second sequence is clear since we have
$$|\frac{1-n}{n}| = |1- \frac{1}{n}| < 1 \quad \forall n > 0$$
