Chapter 5.4, Problem 238E

Use the limit comparison test to determine whether each of the following series converges or diverges.

238. Complete the details of the following argument: If ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n}}$ converges to a finite sum s. then

$\frac{1}{2}s=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\mathrm{...}$ and $s-\frac{1}{2}s=1+\frac{1}{3}+\frac{1}{5}+\mathrm{...}$

Why does this lead to a contradiction?