What is the contradiction in the proof of Hahn Decomposition Theorem?

[nateHI]

In the proof of the Hahn Decomposition Thm located here, there is the following sentence at the top of page 4.

"However, we shall prove that there is a positive subset of E carrying a positive charge, thereby obtaining the
contradiction."

but, if $G$ is the positive subset of $E$ mentioned above then
$G \subset E\subset N$ which implies that $N$ is not a negative set. That seems to support the incorrect assumption, not contradict it. What am I missing?

 

[mathman]

I haven't gone through the details, but I think it is trying to show that $G\subset{P}$ contradicting $G\subset{N}$.

 

[nateHI]

I haven't gone through the details, but I think it is trying to show that $G\subset{P}$ contradicting $G\subset{N}$.

Yup. You're correct. Thanks, I was having a hard time with that one.