I have the following exercise:

Prove by induction the truth of the following statement:

P(n) : 3

^{n}< (n+2)! (for all n in N)

The statement P(n) is true for n=0:

3

^{0}< (0+2)!

1 < 2

Assuming P(n) true for a particular value of n, from the truth of P(n) we will try to prove the truth of P(n+1).

Multiplying both members of P(n) by 3, we obtain:

3

^{n}· 3 < 3(n+2)!

3

^{n+1}< 3(n+2)!

3

^{n+1}< (n+3)(n+2)! < (n+3)!

we have written (n+3) in the place of 3. And also:

n+3 > 3, therefore the direction of the inequality remains the same.

And also, since n!=n(n-1)! we have (n+3)! = (n+3)(n+2)!

In conclusion:

3

^{n+1}< [(n+1)+2]!

This last statement confirm that P(n) is valid also for n+1 and therefore, by the principle of mathematical induction, P(n) is valid for all n in N

END.

I have some doubts here, and I have some questions to ask:

n+3 > 3, therefore the direction of the inequality remains the same.

why is it valid that substitution?

If we are considering by inductive hypothesis that n≥0,

considering n+3 > 3, and since n≥0, let's take for example n=0: substituting we obtain 3>3 that it is false!! why?

please, can you explain me betters this matter? Many thanks!