Ratio Test question (proof)

Ratio Test question (proof)

In the proof of the Ratio Test. We assume the terms of the sum are all
positive and we have $$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L < r <
1$$
Then we say there is an $N$ such that $$\frac{a_{n+1}}{a_n} \leq r$$
when $n \geq N$
Why are we allow to have $\leq$? I am under the impression that this is
what's going on
Since $\frac{a_{n+1}}{a_n} $ converges, then for any $\epsilon >0$, there
is an $N$ such that $\forall n \geq N$, we have $$\left |
\frac{a_{n+1}}{a_n} - L\right | < \epsilon \iff -\epsilon + L <
\frac{a_{n+1}}{a_n} < L + \epsilon$$
Here we take $r = L + \epsilon$ and we are only looking at the right
inequality.
Also, can someone write me a formula for a sequence that has a limit $L$,
but goes over $1$ initially (or sometimes) and then goes near $L$ after a
very long time (for large $N$)?