On the ratio of binomial and Poisson probability distributions (Q1382445)

Consider the binomial distribution \[ {\mathcal B} (i;n,a)= {n\choose i} a^i(1-a)^{n-i} \quad \text{for} \quad i=0,1, \dots, n. \] Consider also the Poisson distribution \({\mathcal P}\) having the mean \(na\) as of \({\mathcal B}\) above, i.e. \[ {\mathcal P} (i;na)= {(na)^i e^{-na} \over i!} \quad \text{for} \quad i=0,1,2, \dots. \] It is shown that for each fixed \(i\in \mathbb{N}\) and for each fixed \(a\in(0,1)\) the inequality \[ \sup_{n\geq i} {{\mathcal B}(n-i;n,a) \over{\mathcal P} (n-i,na)} \leq \sup_n {{\mathcal B} (n;n,a) \over {\mathcal P} (n,na)} \] holds.