<s>[INST] For $0 \leq a,b \leq 1$, why does $aFor $0 \leq a,b \leq 1$, why does $a<b$ imply $\frac a{1-a }< \frac b{1-b}$?
Do we need to compute derivatives to validate this inequality? Or talk about increasing/decreasing functions?
 [/INST] Rearrange the inequality as follows:
\begin{align}&\frac{a}{1-a}<\frac{b}{1-b}\\&\frac{a}{a-1}>\frac{b}{b-1}\\&\frac{a-1+1}{a-1}>\frac{b-1+1}{b-1}\\&1+\frac{1}{a-1}>1+\frac{1}{b-1}\\&\frac{1}{a-1}>\frac{1}{b-1}\end{align}
From $a<b$ it follows that $a-1<b-1<0$ and the inequality is thus satisfied.
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