Hey everyone. I have a question on the definition of limits basically. First, the definition states:

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Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say taht the limit of f(x) as x approaches a is L, and we write

if for every number

there is a corresponding number

such that

if

then

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So the definition is basically saying that if the difference between x -a (whether x is approaching from the left/right since its abs value bar) is approaching 0, then the difference between f(x) - L is approaching 0. Is that correct?

Now an example:

Prove that

Solution:

So we say: if

then

And that's the end of it. I don't see how that proves anything besides that delta and epsilon are proportional. How does this prove anything?