Let X$X$ be a set of sequences of real numbers that converge to zero with the property that forfor any unbounded sequence of real numbers (y_n)$(y_n)$, there is a sequence (x_n)$(x_n)$ in X$X$ for which the coordinatewise product (x_ny_n)$(x_ny_n)$ is unbounded. That is, the sequence (x_n)$(x_n)$ detects the unboundedness of (y_n)$(y_n)$. Such a set X$X$ has a uniform boundedness property: any sequence (y_n) for$(y_n)$ for which (x_ny_n)$(x_ny_n)$ is bounded for all (x_n)$(x_n)$ in X$X$ must be bounded. My
My questions are: what is the smallest possible cardinality of the set X$X$? Also, if we take the linear span of X$X$, what is the smallest possible algebraic dimension of span(X)$\text{span}(X)$?
