We have a box containing red and black balls. If we draw two at random the probability of getting both of them red is $1/2$. Which basically means:
\begin{equation} \frac{r}{r+b} \cdot \frac{r-1}{r+b-1} = \frac{1}{2} \end{equation} Then we have for a positive number of red and black balls, $r$ and $b$ respectively:
\begin{equation} \frac{r}{r+b} > \frac{r-1}{r+b-1} \end{equation}
and the following inequality follows:
\begin{equation} \left(\frac{r}{r+b}\right)^2 > \frac{1}{2}>\left(\frac{r-1}{r+b-1}\right)^2 \end{equation}
How can I derive this inequality?