Hi, so I need to prove that 2^n > n for all positive numbers. I'm not sure if this is the right place to post this (couldnt find anywhere for basic proofs, but this is for my college linear algebra/intro proof class).

My attempt:

2^(n+1) > n + 1

Step 1:Since 2^(n+1) > n + 1 if 2^(n + 1) > 2(n+1) then 2^(n+1) > n + 1

Step 2: 2^(n+1) > 2(n+1)

Step 3: 2^n * 2 >2(n+1)

Step 4: 2^n > n + 1

Step 5: Since 2^n > n + 1. 2^n will be greater than n.

Is this a good way to prove this?? I Feel like it's wrong/terrible verbose. Any suggestions or better ways to do so (without having to to do all the replacements)?