**Edit:** This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its own post here.

I would not be particularly surprised if the inequalities I want are readily available in several standard texts. Unfortunately, these days all of my probability books are stuck in my office (and I'm stuck at home). So thanks for the help in advance.

Let $B_{n,p}$ denote the usual binomial random variable (i.e., the probability that it equals $k$ is given by ${n \choose k} p^k (1-p)^{n-k}$). I would like some references (or short proofs) for the following fact:

- For all integers $n, k$, and all $0 < p < 1$, we have $\mathbb{P}(B_{n,p} = k) - \mathbb{P}(B_{n,p} = k+1) \leq \dfrac{100}{n p (1-p)}$

[I'd be happy if the number "100" is replaced by whatever universal constant is convenient.]

I was having trouble coming up with a particularly good proof of this, so that would be welcome. But ideally, I would prefer a reference if possible. Thanks!

(If curious, this claim *could* be proven by looking at the left-hand-side as a function of $k$, noting when it's increasing [e.g., by taking consecutive differences], and checking the value at this max. Unsurprisingly, this is maximized when $k$ is one standard-deviation above the mean [this corresponds to the inflection point in the normal distribution])