<s>[INST] Proving $\frac{x+1}{y+1}$ is equal to or less than $\frac{x-1}{y-1}$ I know that $\frac{x+1}{y+1}$ is equal to (when $x = y$) or less than $\frac{x-1}{y-1}$. Suppose we cross multiply, then


*

*$xy - x+ y -1$ is less than or equal to $xy + x - y -1$
So,

*$y- x$ is less than or equal to $x- y$.
But when $y$ is greater than $x$ then the inequality is wrong.
Where am I wrong?

 [/INST] If $ 1 \leq y \leq x $ then $ y-x \leq x-y $. Now add $xy-1$ to both sides and factorise to get 
\begin{eqnarray*}
\underbrace{xy+y-x-1}_{(x+1)(y-1)} \leq \underbrace{xy+x-y-1}_{ (x-1)(y+1)} \\
\frac{x+1}{y+1} \leq \frac{x-1}{y-1}.
\end{eqnarray*}
</s>