The Hoeffding's inequality is $P(S_n - E[S_n] \geq \epsilon) \leq e^{-2\epsilon^2/k'}$, where $S_n = \sum_{i=1}^{n} X_i$, $X_i$'s are independent bounded random variables, and $k'$ depends on the random variables. In the proof of Hoeffding's inequality, an optimization problem of the form is solved: $$\min_{s} \ \ e^{-s\epsilon}e^{ks^2}$$ subject to $s > 0$, to obtain a tight upper bound (which in turn yields the Hoeffding's inequality). It turns out that $s = \epsilon/2k$ is the value that obtains the Hoeffding's inequality. I don't see how.
EDIT: Note that $k > 0, \epsilon > 0$.