Is the simpler proof for Descartes' Rule of Signs valid?

[Millennial]

People who are actually supposed to answer this question are those who know about the Descartes' Rule of Signs, so I will not go about explaining it. The well-known proof for the Rule includes somewhat 6 lemmas and covers 7 pages or so, presented http://homepage.smc.edu/kennedy_john/POLYTHEOREMS.PDF , starting from (17). However, I came across a simpler proof that is presented http://www.math.tamu.edu/~rojas/wangdescartes.pdf , which covers somewhat 2 pages. My question is that is this last proof valid? I did not spot any mistakes so far, I am curious if you will.

 

[AlephZero]

The big difference is that the first proof is rigorous and uses only algebra, but the second one uses properties of continuous functioms which the author claims are "obvious".

They are "obvious" in the sense that you can draw some pictures to show they are plausble, but that isn't a rigorous proof. For example it is easy to invent a continuous function p(x) with p(0) > 0 and $p(x) \rightarrow \infty$ as $x \rightarrow \infty$, which crosses the positive x-axis an infinite number of times. He doesn't attempt to prove that such a function can not be a polynomial.

 

[Millennial]

Maybe, but a function that has finite maxima/minima is bound to have finite x-intercepts. Polynomials have finite maxima/minima because their derivatives are also polynomials, and a polynomial of a finite degree has finite solutions.