How to prove this kind of statement: if $a$, then $b$ or $c$.

Question

I think I saw a strategy to prove this statement is to suppose $a$ and $b$ are true then to prove that $c$ is false. Is it correct?

Answer 1

No. If you suppose $a$ and $b$ are both true, then you've already assumed what you need to prove, since under those circumstances we have $b$ or $c$ being true anyway.

You must assume $a$ and then show that either one of $b$ or $c$ is true; or you must assume $b$ and $c$ are both false, and show that $a$ is false. One strategy for the former, as pointed out by @Clayton in the comments: show that if $a$ is true and $b$ is false, then $c$ is true. One strategy for the latter: assume that $a$ is true but $b$ and $c$ are both false, and then derive a contradiction.