TITLE: How to check whether the following inequality holds or not?
QUESTION [7 upvotes]: I am given with real numbers $r,k > 1$. Is the following true? $$\frac{rk}{r+k-1} > 1$$ I have taken a few reals and the result holds for them. Does this inequality holds in general or not?

REPLY [5 votes]: $0 \lt \cfrac{1}{r} \lt 1\,$ and $0 \lt \cfrac{1}{k} \lt 1\,$ since $r,k > 1$, therefore:
$$\frac{rk}{r+k-1} = \frac{1}{\cfrac{1}{r}+\cfrac{1}{k}-\cfrac{1}{rk}} = \frac{1}{1 - \left(1-\cfrac{1}{r}\right)\left(1-\cfrac{1}{k}\right)} > \frac{1}{1} = 1$$