In the textbook Algorithms (4th edition, by Robert Sedgewick and Kevin Wayne), page 466 says:
for the classical Poisson distribution: we assume variable k has a pdf $$\frac{\alpha ^{k}e^{-\alpha }}{k!},$$
then we have the following probability inequality to estimate the upper bound of right tail:
$$Prob(k>t\alpha )<\left (\frac{\alpha e}{t} \right )^t e^{-\alpha } .$$
How does one prove this inequality?