TITLE: Good upper bound for $(1-x)^r$
QUESTION [9 upvotes]: The Bernoulli's inequality gives a lower bound on numbers of the form $(1-x)^r$:
$$(1-x)^r\geq 1-rx$$ for integer $r\geq 0$ and real number $0<x<1$.
Is there a corresponding upper bound for $(1-x)^r$? In particular, when $r$ gets large, $(1-x)^r$ becomes very small. I suspect there should be a good bound for it, but I don't know of any.

REPLY [5 votes]: The following inequality may be easier to deal with depending on what you're trying to use it for
$$(1-x)^r \leq e^{-xr}.$$
The reason is that $1 + x \leq e^{x}$ for all real $x$. (Note that the $r \geq 0$ assumption is crucial for this inequality to be true.)