[Math] Let $n$ be a positive integer, Prove that $\sum_{k=1}^n\frac{ (-1)^{k-1}}{k}{n \choose k} = H_n$

binomial-coefficientscombinatorial-proofscombinatoricsproof-explanation

Let $n$ be a positive integer, Prove that $\sum_{k=1}^n \frac{(-1)^{(k-1)}}{ k} {n \choose k} = H_n = 1+\frac{1}{2}+\ldots+\frac{1}{n}.$

This problem was solved as an example in Titu Andreescu's Combinatorics book. However I do not understand the last part of the solution.
Here is the solution given by him
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I was able to understand the solution presented above, however. What I do not understand in this

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For those who might want to ask what theorem 3.2(i) states, here it is
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Please someone should help me demystify this solution, Thanks in advance.

I'm not too good with Mathjax so here is the screenshot of the original problem.

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Best Answer

He shows that $$ \sum_{k=0}^{m+1}\left(-1\right)^{k}\binom{m+1}{k}=0 $$ He uses the following property $(i)$ from the theorem $3.2$

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