Let $n$ and $1\leq k \leq n$ be natural numbers. Prove the inequality $$\sum_{i=k}^n \binom{n}{i}\bigg(\frac{k}{n+1}\bigg)^i\bigg(1-\frac{k}{n+1}\bigg)^{n-i} \leq 1 - \frac{1}{e} $$
Equivalently, if $X\sim$ Bin($n$,$\frac{k}{n+1}$), prove that $\mathbb{P}[X\geq k] \leq 1 - \frac{1}{e}$.
My attempt: It may be helpful to show that the LHS tends to a limit less than $1-\frac{1}{e}$ as $n \to \infty$ (already did that) and that the LHS is an increasing function on $n$ (have not done that).
Not a complete answer, but here are few hints:
1) $\binom{n}{k} \leq \frac{n^k}{k!}$
2) $\sum_{k=0}^{n}\frac{z^k}{k!} < e^z \ \text{if} \ z>0$
3) $\sum_{i=k}^{n} p(k) = 1 - \sum_{i=0}^{k-1}p(k)$
Hope this helps