Consider the set of all real sequences (different from the zero sequence) that have the value zero on their elements. We can classify this set in two subsets: (1) one with finite number of zeros and (2) one with infinite number of zeros.
My question is: How one can find a property or a criterion that gives a clear distinction between the two mentioned classes of sequences.
The set ${\cal S}$ of all real-valued sequences ${\bf x}=(x_n)_{n\geq0}$ is divided into two parts ${\cal A}$ and ${\cal B}$. A sequence ${\bf x}\in {\cal S}$ is in ${\cal A}$ if there is an $N\in{\mathbb N}$ such that $x_n\ne0$ for all $n>N$, and is in ${\cal B}$ otherwise.
Note that this "criterion" just repeats more or less verbatim the definition of the distinction ${\cal A}$ vs. ${\cal B}$ you gave in your question. There is not more to say.