TITLE: question on prove by induction that for each n$\in\mathbb{N}_{\ge2}$, $n^2$< $n^3$
QUESTION [0 upvotes]: I have to prove by induction that for each n$\in\mathbb{N}_{\ge2}$, $n^2$< $n^3$.
If I try to prove for P(1) I end up with 1 < 1. Is this right? Why does it or does it not make sense?

REPLY [1 votes]: Technically, you can start at $1$ for this sort of proof. Namely, you are proving:
$$\forall n\in\mathbb N(n\geq 2\implies n^3> n^2)$$
So you prove that since $1\geq 2$ is not true, then $1\geq 2\implies 1^3> 1^2$ is true.
Then show that if the statement is true for $n$, then, if $n+1=2$, prove that case, separately, and if $n+1>2$ then $n\geq 2$, so you can use standard induction, because you can assume $n^3>n^2$ when $n\geq 2$.
If $P(n)$ is a statement, and $m$ is a natural number, in order to prove:
$$\forall n(n\geq m\implies P(n))$$
You can prove: $P(m)$ and $$\forall n (n\geq m\implies (P(n)\implies P(n+1))$$
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