Because you need N to be large enough that both and are within epsilon of their limits. You can chuse any both and , the max is just the smallest one that will work.
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.