Prove 2^n > n for all n equal to or greater than 1. i mostly need with how to solve the problem when it is greater than rather than equal to. for the

If is an integer, you can use induction. First show the inequality holds for . You have , which is true.

Now assume this holds in general for , i.e. that . We want to prove the statement then must hold for .

Because , you have



and this must be greater than for the statement to be true, so we require



for . Well this is obviously true, because solving the inequality gives . So you're done.

If you is any real number, you can use derivatives to show that increases monotonically and faster than .