Prove the existence of two constant to bound a function

Question

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a continous function, such as exists $p\in\mathbb{R}^+$, $p>1$ verifying: $$\liminf_{x\to\infty}\frac{f(x)}{|x|^p}=L\in(0,+\infty]$$

Prove the existence of two constants, $r,m\in\mathbb{R}^+$ for which $f$ verifies:

$$f(x)\geq mx^p \;\;\;\; \;\; \forall x > r$$

Any hints about how to approach this exercise?

Answer 1

For any $m \in (0,L)$ we have $lim \inf \frac {f{(x)}} {|x|^{p}} >m$. This implies there exist $r$ such that $f(x) >mx^{p}$ for all $x \geq r$.