Let $p:\mathbb{R}\rightarrow\mathbb{R}$ be a polynomial of odd degree. Prove that there is a solution of the equation $$p(x)=0, x\in\mathbb{R}$$
I am giving this question in an analysis textbook and the only machinery I have to work with is the sequential definition of continuity and the intermediate value theorem. Using only these ideas I am having difficulty coming up with a concrete proof.
Hint: if $p$ is a polynomial of odd degree, then $$\lim_{x\to +\infty}p(x)=\pm \infty, \quad \lim_{x\to -\infty}p(x)=\mp\infty $$