N Choose k is not greater than n^k/Proof 1

Proof
First let $k > 1$.

For $k > 1$, the product has at least two factors.

Hence at least one factor is strictly less than $n$.

Let $k < 0$.

By definition of binomial coefficient:


 * $\dbinom n k = 0$

while $n^k > 0$.

Hence the result follows for $k < 0$.

Finally let $0 \le k \le 1$.

From Binomial Coefficient with Zero:
 * $\dbinom n 0 = 1 = n^0$

From Binomial Coefficient with One:
 * $\dbinom n 1 = n = n^1$

So equality holds when $k = 0$ or $k = 1$.

Hence the result has been shown to hold for all $k \in \Z$.