Not at all.

Once you proved a theorem through delta-epsilon. You can prove another theorem about limits without relying on on delta-epsilon. In fact, later on delta-epsilon is no longer used because all the important theorem were established by delta-epsilon.

Here is an example, for prove that,

If you are going to use delta-epsilon I would love to see how suffer trying to prove it.

However, you can do this.

We note that, if

Then,

Were,

and

Since,

And,

Thus,

I just used the limit composition rule.

Its proof is based on delta-epsilon.

But this prove did not have to be.

See what I am saying.

---

Analogy, most theorems are not proven by set theory. They are proven by other theorems proven by set theory.