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I am really having trouble seeing why

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

$$0 < y_1 < 1, \space 0<y_2<1$$

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?

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  • $\begingroup$ The above is valid in the same way that $0<1<2<3$ implies $0<2<3$ $\endgroup$ – Ishan Deo May 16 '19 at 7:10
  • $\begingroup$ That does make sense. Why is it that people see this? don't know . . . $\endgroup$ – hyg17 May 16 '19 at 7:12
  • $\begingroup$ 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. $\endgroup$ – Ishan Deo May 16 '19 at 7:13
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From $0<x_2<1$ and $y_2=x_2$ we get $0<y_2<1.$

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

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  • $\begingroup$ THANK YOU!!! Finally, something that I understand. One quick question if you do not mind, though. Would my last expression in the question be invalid? $\endgroup$ – hyg17 May 16 '19 at 7:13

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