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
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.