Is this proof of the arc length formula valid?

Question

The author uses only the definition of the derivative to prove the arc length formula. He does not invoke the Mean Value Theorem. Is this a valid proof? He claims that the difference quotient over each subinterval will approach the derivative of $f$ at $x_{i-1}$ as $x_i$ approaches $x_{i-1}$. Arc length proof without MVT

Answer 1

It is not a valid proof because he doesn't show why the sum actually converges to that Riemann integral. The reason is that by using the mean value theorem, the sum takes the form of a Riemann sum. Since the width of the largest subinterval of the partition goes to $0$ as $n \to \infty$, the Riemann sum converges to the integral (it is a theorem that $f \colon [a, b] \to \mathbb{R}$ is Darboux integrable if and only if for any sequence of tagged partitions whose maximum subinterval width goes to $0$, the Riemann sums associated to those partitions converge to a single number). This is all assuming that $f'(x)$ is Riemann integrable of course.