Growth Bound for a Polynomial Function on the Real Line

Let $\mathbb{R}$ be the real numbers, an ordered field with order $\le$, and write $|t|$ for the absolute value of $t\in\mathbb{R}$. Let $\mathbb{N}$ be the set of natural numbers, with the order also written $\le$, and let powers be the natural number powers of $\mathbb{R}$.

Then the following hold.

1. (Monotonicity in the exponent) Let $m,n\in\mathbb{N}$ with $m\le n$ and let $t\in\mathbb{R}$ with $1\le t$. Then $t^{m}\le t^{n}$.

2. (Polynomial growth) Let $p$ be a polynomial function on $\mathbb{R}$. Then there are $N\in\mathbb{N}$ and $C\in\mathbb{R}$ with $0\le C$ such that

$$|p(t)|\le C\,t^{N}\qquad\text{for every }t\in\mathbb{R}\text{ with }1\le t.$$