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

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

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