How to check whether the following inequality holds or not?

The inequality is trivial if you write $a := r-1>0$, $b := k-1>0$. Namely $$ \frac{rk}{r+k-1} = \frac{(a+1)(b+1)}{a+b+1} = \frac{ab +a+b+1}{a+b+1} > 1. $$

$$\frac{rk}{r+k-1}>1 \leftrightarrow \\ rk>r + k -1 \leftrightarrow \\ rk -r>k-1 \leftrightarrow \\ r(k-1)>k-1\leftrightarrow\\ r>1$$ These equations are equivalent. Note that the reason we can say this is: $r+k-1>0$ and $k-1>0$. So yes, it holds for every $k,r>1$

$0 \lt \cfrac{1}{r} \lt 1\,$ and $0 \lt \cfrac{1}{k} \lt 1\,$ since $r,k > 1$, therefore:

$$\frac{rk}{r+k-1} = \frac{1}{\cfrac{1}{r}+\cfrac{1}{k}-\cfrac{1}{rk}} = \frac{1}{1 - \left(1-\cfrac{1}{r}\right)\left(1-\cfrac{1}{k}\right)} > \frac{1}{1} = 1$$