I thought I understood how to show that a function has a limit using the limit definition, but my book is doing a problem in a way I can't follow. The question is "prove that the limit of x^2 as x approaches 2 is 4"

The book starts with the epsilon equation

and they show that

which I can understand (though somehow they assume epsilon is less than 4)

Then they say that they want to find delta. They show that delta is

Which I am now able to follow although it took awhile to see why this is true. Up to this point I can understand what is going on. This is all the "scratch work" if I am correct. Now they have to actually prove the limit and they say the following...

"If delta has this or any smaller positive value, the inequality

0<abs(x-2)<delta

will place x between

This completes the proof."

I don't understand how this completes the proof. I can prove the limit a different way but it is disheartening that I can't follow this. Can anyone explain how their work proves the limit. Thanks.