Do there exist proofs of the major limit theorems for sequences that don't involve picking magic epsilons? I call such epsilons magic because they seem to appear in the proof out of thin air and indeed they work, but where they come from is unknown. Yes, I know in how to work backwards in these sorts of proofs, in general, but rather than struggling through the tedious process of picking epsilons and juggling inequalities - which I find less than enlightening - I would rather learn how to provide conceptual/intrinsic proofs for such things.
For instance, consider the standard proof for showing that the limit of a sequence, if it exists, is unique; it's a very basic $\frac{\epsilon}{2}$ argument but it, indeed, contains epsilons. On the other hand, in any Hausdorff space one can show almost immediately that the limit of a sequence must be unique because: (1) Any two distinct points can be separated by open sets (2) Any open neighborhood of a limit point must contain infinitely many terms of the sequence (3) Ergo, only one neighborhood can contain a (the) limit. I find this argument much more compelling, meaningful and instructive than the $\epsilon$-based argument.
With this background in mind, I have the following specific questions: Given two sequences of real numbers $(x_n)$ and $(y_n)$ that converge to real numbers $x$ and $y$ respectively:
1) How does one show that $(x_ny_n) \rightarrow x \cdot y$ without invoking $\epsilon$-based arguments?
And, assuming that $y \neq 0$,
2) How does one show that $\frac{x_n}{y_n} \rightarrow \frac{x}{y}$ without invoking $\epsilon$-based arguments? ?