Proving n^3>n^2: Using Mathematical Induction and Set of Natural Numbers

[Ed Quanta]

How do you go about proving a statement like n^3>n^2 for n is equal to or greater than 2? I can prove this using mathematical induction, but I am unsure how to show n^3-n^2 is an element of the set of natural numbers without just saying in general that this must always yield a number equal to or greater than 1.

 

[suyver]

Can't you just say that because

\forall\alpha>1:\alpha n^2 > n^2

we must conclude that

\forall n>1: n^3 > n^2

 

[HallsofIvy]

Why not just note that n³- n²= n²(n-1)? As long as n> 1, this will be a positive integer.

 

[Ed Quanta]

Yeah, Halls of Ivy, that's what I was thinking. It just looked like too short a proof so I thought I was missing something. Thanks dudes.