I got this proof off this site: Squeeze Theorem - ProofWiki

I've got a question about the step where it says Let N= max{N1,N2}. Why is it needed? and what is it saying? because I've got the same proof in a book which says in the last step that

L-e < f(x) =< g(x) =< h(x) < L+e whenever x> max{N1,N2}

This is the proof I'm referring to from the website:

Note from the corollary to Negative of Absolute Value, we have .

Let .

We need to prove that .

As we know that .

As we know that .

Let .

Then if , and .

So:

- ;
- .

But .

So

and so .

So .

Hence the result.