# What does $0 < x_1 < x_2 < 1$ really mean when $y_1 = x_1/x_2$ and $y_2 = x_2$?

I am really having trouble seeing why

$$0 < x_1 < x_2 < 1$$ is equaivalent to

$$0 < y_1 < 1, \space 0

when $$y_1 = x_1/x_2, \space y_2 = x_2$$.

The part that I am having trouble seeing is that

$$0 < y_1y_2 < y_2 < 1$$ is equivalent to

$$0 < y_2 <1$$

I really want to say

$$0 < y_1 < 1 < \frac{1}{y_2}$$

therefore

$$\infty > \frac{1}{y_1} > 1 > y_2$$

Therefore $$y_2 < \frac{1}{y_1}$$.

Can someone explain this to me?

• The above is valid in the same way that $0<1<2<3$ implies $0<2<3$ – Ishan Deo May 16 '19 at 7:10
• That does make sense. Why is it that people see this? don't know . . . – hyg17 May 16 '19 at 7:12
• Don't worry if you can't see this immediately. Some can, and the others learn to do so by doing many such problems like these. – Ishan Deo May 16 '19 at 7:13

From $$0 and $$y_2=x_2$$ we get $$0
Since $$x_1>0$$ and $$x_2>0$$ we have that $$\frac{x_1}{x_2}>0.$$ From $$x_1 we derive $$\frac{x_1}{x_2}<1.$$