Big O Proof

Hi all! I'm supposed to prove the following:

Suppose that m is a positive real number. Show that Sigma(j^m), j runs from 1 to n is O(n^(m+1)).

So we have:

Sigma(j^m), j runs from 1 to n = 1^m + 2^m + 3^m + . . . + n^m

I think we should use induction on m here to prove this.

Basis Step:

m = 1

This is definitely true for m = 1 because we have

Sigma(j^1), j runs from 1 to n = n(n+1)/2 which is O(n^2)

Now, O(n^2) = O(n^(1+1) = O(n^(m+1)

Inductive Step:

Assume Sigma(j^m), j runs from 1 to n is O(n^(m+1)) to be true. That is, it is true for m = n. (Inductive Hypothesis)

Then must show it is also true for m = n + 1:

Sigma(j^m), j runs from 1 to n + 1

= Sigma(j^m), j runs from 1 to n + j^(n+1)

= O(n^(m+1)) + j^(n+1) (By Inductive Hypothesis)

= . . .

At this point how do I conclude that the whole thing is O(n^(m+1))?

Any help will be appreciated. I'm trying to get a hold of these big O proofs.