Confusion regarding differences between strong induction and simple induction

Question

I don't know how to prove that any proof by induction is also proof by strong induction nor any proof by strong induction can be converted into a proof by simple induction?

An example would be useful in helping me understand!

Answer 1

For simple induction, we show that $P(n-1)\implies P(n)$. For strong induction, we show that $P(1),P(2),\ldots,P(n-1)\implies P(n)$.

Every simple induction proof is trivially a strong induction proof as well.

Given a strong induction proof, we can rephrase it as a simple induction proof by changing the statement being proven from $P(n)$ to the statement $P'(n)$ which says the following: $$ P'(n)=\{P(1),P(2),\ldots,P(n)\text{ are all true}\} $$