I'm having trouble understanding this proof. I understand what the e-d definition is trying to say, and I understand the geometric argument, but... well, consider an example:

Prove that

So, we take the definition and plug in:

For

So, at this point, we have

.

So what we're saying is:

If the distance between x and 3 is less than

, then the distance between f(x) and 7 is less than e.

So? If e is 100, and if |x-3| < 25, then |f(x)-7| < 100.

If e is 10, and if |x-3| < 2.5, then |f(x)-7| < 10.

I can see intuitively that as x gets closer to 3, so does f(x) close in on 7, but it's exactly the same kind of reasoning we used in our informal idea of limits.

I dont' see how this is any more rigorous?

Can I just be mindless about this and find an equivalent statement for the definition (with values plugged in) that is easy to prove? Is that basically the method?