Question about the application of the induction principle

Question

If I want to prove by induction on $n$ that $n! \geq \frac{3^{n-1}}{2}$, I end up with the inequality $n \geq 2$. Is it formally OK if my base is $n = 2$, and if prove that the proposition is true for all $n \geq 2$, and if, after proving that p(1) is true, I assert that p is true for all $n \in \mathbb{N}$?

Answer 1

Yes, that is fine: If you prove that P holds for for all $n \geq 2$, and also that P holds for $n=1$, you have proven P for all $n\in\Bbb N$.

I'm a bit leary, though, when you say "I get $n\geq 2$". Did you really mean to say that your induction proof needed to use $n\geq 2$ for the induction step (and therefore was only valid starting at $n=2$)?

Finally, consider removing the tag "abstract-algebra" from this question; it has nothing to do with that topic.

Answer 2

You could always define an "adapter" clause $P'(n) = P(n+1)$, and do your bog-standard inductive proof on $P'$. Then use your proof of $P(1)$ and your inductive proof of $\forall n \ge 1 P'(n)$ to prove $\forall n \ge 1 P(n)$.

I think this kind of adaptation is seen as "obvious" and is generally implicitly accepted as valid. But if you want to be formally explicit, spell out your use of $P'$.