Short proof of Seidel-Stern theorem on continued fractions

Question

Let $\mathbf{a}=\{a_n:n\ge0\}$ be a sequence of positive real numbers, and consider the formal continued fraction $$K(\mathbf{a})=a_1+\cfrac{1}{a_1+\cfrac{1}{a_2+\ddots.}}$$

Seidel-Stern Theorem. If $\sum_{n\ge0} a_n=\infty$, then the formal continued fraction $K(\mathbf{a})$ converges.

Is there a direct/short proof of this theorem?

Thanks!

Answer 1

You can find the proof in the book Continued Fractions by A. Ya. Khinchin (University of Chicago press, 1964). It is theorem 10 on page 10.