Can anyone provide a hint as to how I can prove the sequence $\sum_{k=1}^{n}\frac{(-1)^k}{k}$, sequence c in the attached question, is Cauchy? This is from Russell A. Gordon's real analysis: a First Course.
The reason I believe this has to be done through Cauchy convergence is because it is from the same section in the book (Cauchy convergence and bounded monotone convergence) and parts a and b are somewhat easily proven to be Cauchy convergent. It also has an oscillatory behavior. Obviously series tests are not applicable either for the same reason, i.e., not encountered yet in the book.
I'm unable to figure out how I can reduce $$\left |x_n-x_m \right |=\left |\sum_{k=m+1}^{n}\frac{(-1)^k}{k}\right |$$ to a tractable greater series to be summed, e.g., through the telescopic formula, as in (likely) parts a and b, or any other convergent series.