when the numerator is less than the denominator

Question

when the numerator is less than the denominator the result is always between 0 and 1? for example if I have a number like x/y where x<y then the result will be between 0 and 1 always? Is there a proof for this?

Answer 1

Assuming that $x$ and $y$ are positive, you have $0<x<y$, so $\frac1y>0$, and $$0\cdot\frac1y<x\cdot\frac1y<y\cdot\frac1y\;,$$ which on simplification becomes

$$0<\frac{x}y<1\;.$$

Answer 2

If $0 < x < y$, then by dividing all three numbers by the positive quantity $y$, you have $$ 0 < \frac{x}{y} < 1. $$