Thread: Proof by Induction

Hi, can someone help me prove this inequality by Induction

The inequality is: n! ≥ 2^n-1 for all natural numbers n

Thank You

1) Define the statement that you'll prove by induction holds for every n:



2) Show Statement(1) is true.

3) ASSUME Statement(k) is true for some k >=1. From that algebracially manipulate it so that you can show that Statment(k+1) must also be true.

It should look like this:







4) 1-3 completes the proof by induction:
You've shown Statement(1) is true by #2,
and, by #3, you've shown that if Statement(k) is true, then Statement(k+1) is true.
Therefore, you've proven Statement(n) is true for all n>=1.

Can you please show me the algebra that occurs at step
"Then ... (insert your algebra here - look at the final result you're trying to get to)

Thanks

I'll give you a huge hint.

You have

You want to manipulate that to show that you also have that

What do you have to do to each side of the original inequality to get to this desired outcome?

You'll use that, since