De la Vallée Poussin’s Theorem

Question

I try to understand De la Vallée Poussin’s Theorem. I don't understand the orange bit:

I really don't see why they need to have same sign. Any help would be appreciated :)

Answer 1

To follow the logic clearly, let us represent the last equation simply as $a=b-c$. From the premise that $|b|>|c|$, we need to show that $a$ and $b$ have the same sign. Consider the two cases $a>0$ and $a<0$ separately.

If $a>0$, then $b-c>0$ and so $b>c$. Therefore $b>0,$ since either $b<0$ and $b<c$ or $b>0$ and $c<b$.

If $a<0$, then $b-c<0$ and so $b<c$. Therefore $b<0$, by the same reasoning as above.