Proof. Let $p_n=\prod_{i=1}^n u_i$ . We have to study the convergence of the sequence $\{p_n\}$ . The sequence $\{p_n\}$ converges to a not null limit iff $\{\log p_n\}$ ($\log$ is restricted to its principal branch) converges to a finite limit. By the Cauchy criterion, this happens iff for every $\epsilon^{\prime}>0$ there exist $N$ such that $\abs{\log p_{n+k} - \log p_n} < \epsilon^{\prime}$ for all $n>N$ and all $k=1,2,\ldots$ , i.e, iff $$\abs{\log \frac{p_{n+k}}{p_n}} = \abs{\log u_{n+1}u_{n+2}\cdots u_{n+k}} < \epsilon^{\prime};$$ as $\log(z)$ is an injective function and continuous at $z=1$ and $\log(1)=0$ this will happen iff for every $\epsilon>0$ $$\abs{u_{n+1}u_{n+2}\cdots u_{n+k}-1} < \epsilon$$ for $n$ greater than $N$ and $k=1,2,\ldots$