Real Analysis sequence limit problem

I suppose that there is a typo in your textbook. When it mentions Cauchy, it should mention Cesàro summation instead, since it implies that\begin{align}\lim_{n\to\infty}\frac{\log\left(\frac{x_n}{x_{n-1}}\right)+\log\left(\frac{x_{n-1}}{x_{n-2}}\right)+\cdots+\log\left(\frac{x_1}{x_0}\right)}n&=\lim_{n\to\infty}\log\left(\frac{x_n}{x_{n-1}}\right)\\&=\log a.\end{align}By the way, it's a nice proof.

What's happenning is that if a sequence $a_n$ tends to a limit, then the sequence of averages $\frac{1}{n}\sum_{k=1}^na_k$ also tends to the same limit. This is Cauchy's first theorem of limits, according to the article you referred to.