showing an inequality not using stirling formula I don't know how to show that
$$
\frac{k^k}{k!}\leq e^{k}
$$
without using Stirling's approximation. I want to show it directly. I guess I need some inequality to achieve this but I don't know.

A somewhat different approach:
In the pmf for a Poisson$(k)$ $P(X=k) = \frac{e^{-k} k^k}{k!}$ but $P(X=k)<1$ (since it's a probability).
The result follows by multiplying by $e^k$.
