Here is one of my favorite, that I learned from A. G. Khovanskii: let $f$ be a univariate rational function with real coefficients. Then, you can think of $f$ as inducing a continuous self-map of $mathbb{RP}^1 cong S^1$, in particular, it has a topological degree, say $[f]$, and if $f$ happens to be a polynomial, it is obvious that $[f]=0$ if $deg(f)$ is even, and that $[f]=pm 1$ if $deg(f)$ is odd (depending on the sign of the main coefficient).
If the decomposition of $f$ in continued fraction is
$$ f=P_0+frac{1}{P_1+frac{1}{P_2+ddots}}$$
Then one can prove easily that $[f]$ is the (finite) sum: $[f]=sum_{i geq 0} (-1)^i[P_i]$.
(Khovanskii himself taught this to high-schoolers in Moscow.)
The interesting connection for me follows: for any real polynomial $P$, the topological degree of the fraction $P'/P$ is clearly the (negative of the) number of real roots of $P$. Thus, the computation formula above applied to $[P'/P]$ allows us to recover Sturm's theorem.
I don't know if it really qualifies as a new proof of the theorem, but it's definitely a different point of view on that proof.
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