The Exponential Function Dominates Every Power

Let $\mathbb{R}$ be the real numbers, an ordered field with order $\le$, and let $\exp$ be the exponential function. Let $\mathbb{N}$ be the set of natural numbers, with successor map $S$ as in that definition, let $\iota_{\mathbb{R}}:\mathbb{N}\to\mathbb{R}$ be the canonical map of $\mathbb{R}$, as in clause 3 of The Real Numbers and Standard Notation (written explicitly here, rather than by the abbreviation of that clause, because natural numbers occur below both as exponents and as elements of $\mathbb{R}$), and let powers be the natural number powers of $\mathbb{R}$. For $t\in\mathbb{R}$ with $t\ne 0$ let $t^{-1}$ denote the multiplicative inverse of $t$.

Let $m\in\mathbb{N}$ and put $\mu=\iota_{\mathbb{R}}(S(m))$. Then $0<\mu$, and

$$t^{m}\exp(-t)\le \mu^{S(m)}\,t^{-1}\qquad\text{for every }t\in\mathbb{R}\text{ with }0<t.$$