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

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    Yes? Is there a reason you think it can't? – 2017-02-02
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    I couldn't personally think of a concrete example to that form, meaning I couldn't try to prove it by induction. Therefore, I don't know if it is possible or not. I'd appreciate it, if you could give an example. – 2017-02-02

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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.

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    A very good and clear explanation. Thank you! – 2017-02-02