Inequality involving binomial cofficients

Question

let $n,k,m,\ell$ be positive integers satisfying $k\le n-m,k\le n-\ell$. I am trying to prove the following inequality: $$X=\frac{\binom{n}{k}-\binom{n-m}{k}}{\binom{n-1}{k-1}-\binom{n-1-\ell}{k-1}} \le \frac{n}{k}\cdot \frac{m}{\ell}$$ It might be that the cofficient $n/k$ is sligntly different, for instance $(n-1)/(k-1)$ or something like that. I have tried to upper bound it using common techniques, for example: $$X=\frac{\binom{n}{k}(\binom{n-m}{k}/\binom{n}{k})}{\binom{n-1}{k-1}(\binom{n-1-\ell}{k-1}/\binom{n-1}{k-1}}$$ $$X=\frac{\binom{n}{k}\frac{(n-m)\dots(n-m-(k-1)}{n(n-1)\dots(n-(k-1))}}{\binom{n-1}{k-1}\frac{(n-1)(n-2)\dots(n-1-(k-1)}{(n-1-\ell)\dots(n-1-\ell-(k-1)}}$$ And now we can use the fact that whenm $0\le i\le k$ we have $$\frac{n}{m}\le \frac{n-i}{n-m-i}\le \frac{n-k}{n-m-k} $$ To conclude: $$ X\le \frac{\binom{n}{k}(1-(1-\frac{m}{n})^k)}{\binom{n-1}{k-1}(1-(1-\frac{\ell}{n-k})^k)}$$ $$=\frac{n}{k}\cdot\frac{1-(1-\frac{m}{n})^k}{1-(1-\frac{\ell}{n-k})^k}$$ However, when I try to upper bound this by using inequalities for $(1-x)^t$ for the nominator and denominator, it seems that the denominator cannot be controled properly. I'm guessing that the inequality used for $(*)$ is a bit too loose and might be possible to improve.