The Exponential Function Dominates Every Polynomial Function

Let $\mathbb{R}$ be the real numbers, an ordered field with order $\le$, write $|t|$ for the absolute value of $t\in\mathbb{R}$ and $t^{-1}$ for the multiplicative inverse of $t\ne 0$, and let $\exp$ be the exponential function. Let $p$ be a polynomial function on $\mathbb{R}$.

Then the following hold.

1. (Decay at $+\infty$) For every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $M\in\mathbb{R}$ with $1\le M$ such that

$$|p(t)|\exp(-t)<\varepsilon\qquad\text{for every }t\in\mathbb{R}\text{ with }M\le t.$$

2. (Decay at $0$ from the right) For every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\delta\in\mathbb{R}$ with $0<\delta$ such that

$$\bigl|p\bigl(s^{-1}\bigr)\bigr|\exp\bigl(-s^{-1}\bigr)<\varepsilon\qquad\text{for every }s\in\mathbb{R}\text{ with }0<s<\delta.$$