A note on bounds for the odds theorem of optimal stopping

The odds theorem gives a unified answer to a class of stopping problems on sequences of independent indicator functions. The success probability of the optimal rule is known to be larger than Re-R, where R defined in the theorem satisfies R≥ 1 in the more interesting case. The following findings strengthen this result by showing that $1/e$ is then a lower bound. Knowing that this is the best possible uniform lower bound motivates this addendum.