Logic: Using mathematical induction to prove a conditional statement

Question

Can a statement of the form $$ \forall a \in \mathbb N, P(a)\implies Q(a)$$ be proved by using mathematical induction?

Answer 1

Any proveable statement of the type $\forall a\in\mathbb N: S(a)$, where $S$ is any logical expression about $a$, can be proven by mathematical induction. In your case, $S(a)$ is equal to $P(a)\implies Q(a)$, there's no reason to think $S$ is anything special.

Naturally, induction may not always be the easiest way to prove the statement but it will always work.