Limit Theorems Without Magic Epsilons 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? ?
 A: An approach that is very effective but not quite elementary is given by nonstandard analysis (NSA). Essentially, epsilontics are delegated to some powerful set-theoretic machinery. There is a post by Terry Tao that explains how NSA does the epsilon management for you: Ultrafilters, nonstandard analysis, and epsilon management. NSA allows you to make heuristic arguments like the following precise: We have $$xy-x_n y_n=(x-x_n)y+x_ny-\big((x_n-x)y_n+x y_n\big)$$
$$xy-x_n y_n=(x-x_n)y+x_ny-(x_n-x)y_n-x y_n.$$
Now if $|x_n-x|$ and $|y_n-y|$ are infinitesimal then the terms $(x-x_n)y$ and $(x_n-x)y$ are infinitesimal and the difference between $x_n y$ and $x y_n$ is infinitesimal, so the whole difference is infinitesimal. To make this precise in NSA, one lets $n$ be an infinite natural number and such numbers actually exist. An excellent introduction to NSA is provided by Robert Goldblatt, Lectures on the Hyperreals. 
However, it shouldn't be to hard to turn the heuristic approach into proper epsilontics either.
A: The answer is yes if the property you want to prove is expressible and true in some topological space without a metric.
But it is definitely no if you work with a metric space and its topology which is defined by $\epsilon$ stuffs.
However you can sometimes use a richer structure. For example the topology of $\mathbb R$ is given by the metric but also by the order. Written only with the axioms of the order of $\mathbb R$ (e.g. see Tarski's axioms), proofs are logically simpler (for example if you try to write a proof is a proof assistant) but not shorter when you do it by hand.
