$\displaystyle Prove 1^k+2^k+...+n^k <= n^(k+1), k is a real number greater than 0 and n is an integer greater than 0.$

1. P(n): $\displaystyle 1^k+2^k+... n^k <= n^k+1 $

2. $\displaystyle P(1) 1^k +2^k +... +1^k <= 1^k+1

1^k<=1^k+1, true$

3. Assume: P(n): $\displaystyle 1^k+2^k+... = n^k <= n^k+1 $

4. Prove(n+1)

$\displaystyle n^k+1 + n^k <= n^k+1

n^k+1+(n+1)^k <= (n+1)^k+1$

// Im stuck on how to manipulate this part of the equation, i guess i need to refresh in algebra. Am in correct up till this point, and any suggestions on how to finish this? Thanks and gretaly appreciated.