Question about Leonard Gillman's proof of the divergence of the Harmonic series. Leonard Gillman (1917 – 2009) was an American mathematician, emeritus professor at the University of Texas, Austin. His proof of the divergence of the Harmonic series appeared in The College Mathematics Journal(March 2002) which you can read it here. Page: 5/13
My problem is that I have hard time understanding the conclusion of the proof, especially the contradiction part.It will be great if somebody helps me understand it. Thank you:)
 A: This is a proof by contradiction.
It starts by supposing $S=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\cdots$ converges to a finite sum (it would then be absolutely convergent since all terms are positive)
and then notes that this is strictly greater than $\frac{1}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{6}+\frac{1}{6}+\cdots$ by making fractions with odd denominators smaller (in fact the difference would be $\frac{1}{2}+\frac{1}{12}+\frac{1}{30}+\cdots$)
and then inserts some brackets $(\frac{1}{2}+\frac{1}{2})+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{6}+\frac{1}{6})+\cdots$ which would give $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots$ and this is $S$ again.
So if $S$ converges to a finite sum then $S \gt S$ strictly.  This is a contradiction given that in fact $S=S$, implying that $S$ does not have a finite sum
A: If you assume that the series converges, because it is a series of positive terms, it converges absolutely. So if you reorder, you should get that same $S$. But the argument in the proof shows a reordering that allows you to argue that the number $S$ satisfies $S>S$. As this is impossible, such number cannot exist and the series does not converge. 
