limits - Proof Verification: Show that every polynomial of odd degree with real coefficients has at least one real root.

Question: Show that every polynomial of odd degree with real coefficients has at least one real root.

Proof: $f(x)$ is of the form:

$f(x) = a_nx^n +a_{n-1}x^{n-1}+.....+a_2x^2 + a_1x +a_0$

$f(x)$ is an odd degree polynomial and is hence continuous on $\mathbb{R}$.

Then, if $f(x)$ has a positive leading coefficient

$\implies \lim_{x \to \infty}f(x)=\infty$ and $\lim_{x \to -\infty}f(x)=-\infty$

$\implies \exists \alpha,\beta \in\mathbb{R}$ with $ \alpha > \beta$ such that:

$ \alpha>0$ and $f(\alpha)>0$ and $\beta <0$ and $f(\beta)<0$

Now, consider the restriction of $f(x)$ onto the interval $I:=[\beta,\alpha]$

The restriction of $f$ on $I$ is also continuous on $I$

Then, since $f(\beta)<0

$ \exists$ a number $c \in (\beta,\alpha)$ such that $f(c) =0$

Can anyone please verify this proof and let me know if it is correct and /or if I'm missing out on something?

Note: I've missed out on the case with negative leading coefficient as I believe that too will be done in a similar fashion should this proof be alright.

Thank you.