Prove that the sequence $\displaystyle (\frac{1}{1+n+n^4})$ converges to 0.

Here's the solution:

Here's my problem: I don't understand why they wrote $\displaystyle N=\frac{1}{\epsilon}$ !

We know that $\displaystyle n+1 > \frac{1}{\epsilon}$

Therefore $\displaystyle n> \frac{1}{\epsilon} -1$

And so we should take: $\displaystyle N= \frac{1}{\epsilon} -1$

Isn't that right???

Also, I don't understand why they first omitted $\displaystyle n^4$ from the denominator when they could omit $\displaystyle n+1$ instead. Because I think $\displaystyle \frac{1}{n^4}$ goes to zero a lot quicker. I appreciate some explanation.