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 and be two sequences of real numbers such that and is bounded. Show that .

My first thought was that since is bounded then there must exist a k such that , and so ..? But the proof was...

=> We have M such that for all n. Let . Then we can find m such that n>m implies . Then for n>m we have 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?