Is this a proper proof by induction?

[yyttr2]

I did this is Art of problem solvings Texor, so sorry about the image. Is this a correct and proper proof of 3^n > n^2?

 

[stapel]

\(\displaystyle \mbox{...then since }\, 3^k\, >\, 2k\, +\, 1\, \mbox{...}\)

What is your justification for the "since" here?

 

[yyttr2]

I added an extra proof at the bottom to show that it was true. The first proof is what I actually had to do, the second one I used to prove the first.

 

[JeffM]

I did this is Art of problem solvings Texor, so sorry about the image. Is this a correct and proper proof of 3^n > n^2?

View attachment 3698

OK Quite creative. Well done on the thinking part.

Several technical deficiencies. First, you should show your lemma (and identify it as such) before your proof. Second, you should probably specify that n is not merely positive but an integer. Third, you are not specifying that k must not be less than the number or numbers for which you have shown the proposition to be true directly.

As a matter of personal style, I do not like the word "assume" to introduce your second step. Your first step has already shown that the set of integers satisfying the proposition to be proved is not empty. In your second step, you are now showing that any element of the set (which is not empty) transfers that proposition to its successor.