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 http://www.proofwiki.org/w/images/ma...85e5a76bd4.png.

Let http://www.proofwiki.org/w/images/ma...4984a23fe5.png.

We need to prove that http://www.proofwiki.org/w/images/ma...70d2974672.png.

As http://www.proofwiki.org/w/images/ma...2f6df7c7cf.png we know that http://www.proofwiki.org/w/images/ma...7285d28db9.png.

As http://www.proofwiki.org/w/images/ma...ebfcc02640.png we know that http://www.proofwiki.org/w/images/ma...5895773163.png.

Let http://www.proofwiki.org/w/images/ma...3adf82f01e.png.

Then if http://www.proofwiki.org/w/images/ma...9574566b34.png, http://www.proofwiki.org/w/images/ma...853e4e09c1.png and http://www.proofwiki.org/w/images/ma...ad3b2b8f63.png.

So:

- http://www.proofwiki.org/w/images/ma...f1dafcbfb6.png;
- http://www.proofwiki.org/w/images/ma...5ec0677a0b.png.

But http://www.proofwiki.org/w/images/ma...8d5dfdd543.png.

So http://www.proofwiki.org/w/images/ma...4fab2ea955.png

and so http://www.proofwiki.org/w/images/ma...f8110a69bc.png.

So http://www.proofwiki.org/w/images/ma...0474b29092.png.

Hence the result.