In general, taking the sequence for example, if $\lim\limits_{n \to \infty}a(n)=0$, we call the sequence $a(n)$ is an infinitesimal.
It's well known that, the product of a finite number of infinitesimals is still an infinitesimal, which can be proven by induction. Suppose that $\lim\limits_{n \to \infty}a_1(n)=\lim\limits_{n \to \infty}a_2(n)=0$. Then according to the rule of the limits product, $\lim\limits_{n \to \infty}[a_1(n)a_2(n)]=0$, which shows that the product of two infinitesimals is an infinitesimal. Thus, by induction, we can generalize the conclusion to the case when a finite number of infinitesimals multiply.
But what about the product of infinitely many infinitesimals? How to define such a product?
Dear Sir @Lorenzo, if what you mean is the one above, it would be fair to say that the example is not valid.
Notice that there are $k$ terms greater than $\dfrac{1}{2}$ in $\{x_n^{(k)}\}_{n=1}^{\infty}$. Then let $k \to \infty$, there are infinitely many terms greater than $\dfrac{1}{2}$. As result, $\prod\limits_{k=1}^{\infty} x_n^{(k)}$ is not the product of infinitely many infinitesimals.