 Solution

Every finitely supported sequence vanishes from some index onward, since a finite subset of the natural numbers has a greatest element. Thus the set is a union over $n<\omega$ of sequences determined by their first $n$ real entries, with zeros afterwards. Each nontrivial piece has cardinal ${\mathfrak{c}}^n=\mathfrak{c}$, so the union has cardinal at most ${\aleph }_0\cdot \mathfrak{c}=\mathfrak{c}$.

Conversely $r\mapsto (r,0,0,\dots )$ is an injection from $\mathbb{R}$. Cantor-Schröder-Bernstein therefore givesNo choice of an arbitrary infinite support is involved; it is the finite support that makes a countable union of finite-dimensional parameter spaces enough.