If I've a expression: $-4<3$ and take the absolute value $|-4|<|3|\implies 4<3$ which is false. So I though that maybe the inequality sign would change. But $|-2|<|3| \implies 2>3$ which is also false.
My problem is that I have the inequality $-x<y$ where $x,y>0$. What will happen if I take the absolute value of it: $|-x|<|y|.$ I though it would be $0<|x|<|y|\implies 0<x<y$.
But from the examples above it seems that this ain't true.
I have a increasing sequence $(x_1>x_2>x_3...$ etc.), $x_1,x_2,...,x_n$ where all $x's$ are positive and a constant $m=1/2$. The inequality: $-x_n<1/2$ is always valid because the $x's$ are positive. But $x_3=1$. So for all $n \geq 3$, $|-x_n|>1/2$ .
Given this context helps in solving my problem in taking the absolute value?
Taking the absolute value on both side we have for all $n>3, |-x_n|>|1/2|>0 \implies x_n>1/2>0$.
Is this correct?