I am trying to prove an inequality that would imply my algorithm satisfies $\epsilon$-differential privacy for $k_i$ being a parameter. The inequality is
$ \left(1-\frac{1}{1+e^{\epsilon/k_i}}\right)^{k_i} < 1 - \frac{1-\frac{1}{2^{k_i}}}{e^\epsilon} , $
for $\epsilon > 0$ and $k_i \in \{1,2,...\}$.
I've been trying for weeks, and I've got a rather large proof. First by showing that the two sides are never equal when $\epsilon > 0$, and using the Intermediate Value theorem to show that if at a given $\epsilon^\prime$ and $k_i^\prime$ the RHS is less than the LHS, then this also holds for all $\epsilon>0$ for the same $k_i^\prime$. After that I show the base case at $k_i=1$ and proceed by induction on $k_i$ to show it holds for all $k_i$.
The resulting proof are rather long (5 pages with lots of hyperbolic tangents and friends). I was just wondering if there is a faster and more elegant way to prove the same result.