Taylor series of exponential function inequality example

Question

Assume $d$ positive integer and $\epsilon$ small positive real. How one infers that $$(1-\epsilon)^d\leq e^{-\epsilon d}$$ using the Taylor expansion of exponential function?

Answer 1

This follows from $1 - \epsilon \le e^{-\epsilon}$, which you can verify using calculus (show $f(x)=e^x - x - 1$ is nonnegative by computing its minimum).