Show that for natural numbers $x$ and $y$ if $\frac{x}{y}>1$ then

${\displaystyle \frac{x}{y}>\frac{(x+1)}{(y+1)}>1.}$

Hence prove that

${\displaystyle P=\frac{2}{1}.\frac{4}{3}.\frac{6}{5}...\frac{k}{k-1}>\sqrt{k+1}.}$

This shows that the product $P=\underset{i=1}{\overset{n}{\Pi }}\frac{2i}{2i-1}$ tends to infinity as $n$ tends to infinity. Now, using a similar method, show that

${\displaystyle Q=\frac{2}{1}.\frac{4}{3}.\frac{6}{5}...\frac{100}{99}>12.}$