Curious inequality with binomial coefficients While dealing with a larger task, I've faced an inequality which I found hard to prove:
$$
\frac{i}{n-i-1} \cdot \frac{
   1 - 1/{n \choose i+1}
}{
   1 - 1/{n \choose i}
} \leq 1
$$
here $n \in \mathbb{N}$ and $i \in \left\{1, \dots, \lfloor n/2\rfloor - 1\right\}$ (or, equivalently, for integer $i\geq 1$ such that $2 i \leq n - 1$). I believe that this inequality holds, because I've checked it numerically for lots of values of $n$ and $i$.
I've tried dealing with both multipliers independently and observed that $\frac{i}{n-i-1}\leq 1$ holds for $2 i \leq n - 1$, but for these values of $i$ $\frac{1 - 1/{n \choose i+1}}{1 - 1/{n \choose i}} \geq 1$, so it doesn't lead to desired result.

Your inequality holds!
Note that $\binom{n}{i+1}=\frac{n-i}{i+1}\binom{n}{i}$ and therefore the inequality is equivalent to
$$i-\frac{i(i+1)}{(n-i)\binom{n}{i}}\leq n-i-1-\frac{n-i-1}{\binom{n}{i}}$$
that is
$$\frac{(n-i)(n-i-1)-i(i+1)}{\binom{n}{i}}\leq (n-2i-1)(n-i)$$
or
$$n(n-2i-1)\leq (n-2i-1)(n-i)\binom{n}{i}.$$
If $n\geq 2i+1$ then it suffices to show
$$n\leq (n-i)\binom{n}{i}=n\binom{n-1}{i}$$
which holds since $\binom{n-1}{i}\geq 1$.
