Divisibility of Numerator of Sum of Sequence of Reciprocals/Lemma

Lemma for Divisibility of Numerator of Sum of Sequence of Reciprocals
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then:
 * $\ds \sum_{k \mathop = 1}^n \dfrac {\paren {-1}^{k - 1} } k \dbinom n k = \sum_{k \mathop = 1}^n \dfrac 1 k$

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

Proof
Expanding the summation:

Let:

Then we have: