Calculus proof methods...

Maybe in the wrong forum...

I was just curious as to how everyone here approaches rigorous calculus proofs, I'm thinking more specifically about convergence and related topics. I find that no matter how many examples I do I always find myself lost on the next Q. Here's an example...

Let $(a_n)$ and $(b_n)$ be two sequences of real numbers such that $a_n -> 0$ and $(b_n)$ is bounded. Show that $a_n b_n -> 0$ .

My first thought was that since $b_n$ is bounded then there must exist a k such that $|b_n| < k$ , and so $|a_n b_n| -> 0k = 0$ ..? But the proof was...

=> We have M such that $|b_n| \le M$ for all n. Let $\epsilon > 0$ . Then we can find m such that n>m implies $|a_n| < \frac{\epsilon}{M}$ . Then for n>m we have $|a_n b_n| \le M|a_n| < M\frac{\epsilon}{M} = \epsilon$ as required.

Is there a way to learn these methods other than doing lots of examples which, i feel aren't really helping me... About to start subsequential limits and every Q scares the hell out o me.

So basically, strange post i know but does anyone have any methods they use for proofs?