Outline proof (I'll do it for real ##f, g## just to simplify the notation a bit):
1) The first thing we must do is state clearly what we are going to prove:
Let ##f, g## be continuous at some point ##a##. Then the product function ##fg## is continuous at ##a##.
2) First technicality: what happens if ##f## and ##g## have different domains? What if ##f(x) = \sqrt x##, with domain ##[0, \infty)##; and, ##g(x) = \sqrt {-x}##, with domain ##(-\infty, 0]##. The product function is only defined at ##x = 0##. That's okay, because then ##fg## is continuous vacuously at ##x = 0##.
3) Do we mention domains further or just gloss over that point? For simplicity, let's assume that we have restricted our attention to a shared domain that inlcudes ##a## and not say any more about this.
4) The key inequality is: $$|f(x)g(x) - f(a)g(a)| \le |f(x)||g(x) - g(a)| + |g(a)||f(x) - f(a)|$$How do we deal with this?
4a) Note that ##|g(a)|## is some fixed number (but it might equal ##0##, so we need to be careful about that). The term ##|f(x) - f(a)|## can be made arbitrarily small. We're okay there.
4b) We can make ##|f(x)|## close to ##|f(a)|##, so we must be able to limit the size of this term. And, the term ##|g(x) - g(a)|## can be made arbitrarily small. We're okay here as well.
4c) We have the sum of two expressions, so we should try to make each ##< \frac \epsilon 2##.
5) Let's look at each of those points in more detail.
5a) If ##|g(a)| \le 1##, then the first term is easy. We can find ##\delta_1## such that ##|f(x) - f(a)| < \frac \epsilon 2##. And, if ##|g(a)| > 1##, then we need to find ##\delta_2## such that ##|f(x) - f(a)| < \frac \epsilon {2|g(a)|}##.
Alternatively, if ##|g(a)| = 0##, then the first term vanishes and otherwise we can find ##\delta_2## such that ##|f(x) - f(a)| < \frac \epsilon {2|g(a)|}##.
Another alternative, we could find ##\delta_3## such that ##|f(x) - f(a)| < \frac \epsilon {2(1+ |g(a)|)}##. That takes care of all cases.
We need to decide on one of those approaches. Pick whatever looks best.
5b) The term ##|f(x)|## is trickier, because ##x## is a variable. But, we can find ##\delta_4## such that ##|f(x)| < |f(a)| + 1##. We should be able to show that from the triangle inequality.
That just leaves ##\delta_5## such that ##|g(x) - g(a)| < \frac \epsilon{2(|f(a)| + 1)}##.
5c) We just need to decide on the best approach, rationalise our deltas, and take ##\delta = min(\delta_1, \delta_2)## and we are good to go.
6) How do all epsilon-delta proofs start?
Let ##\epsilon > 0 \dots##