Increasing Alternating Sum of Binomial Coefficients

Theorem
Let $$n \in \Z$$ be an integer.

Then:
 * $$\sum_{j=0}^n \left({-1}\right)^{n+1} j \binom n j = 0$$

where $$\binom n k$$ denotes a binomial coefficient.

That is:
 * $$1 \binom n 1 - 2 \binom n 2 + 3 \binom n 3 - \cdots + \left({-1}\right)^{n+1} n \binom n n = 0$$

Proof
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