= 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 $r\mapsto(r,0,0,\ldots)$ is an <injection> from $\mathbb R$. Cantor-Schröder-Bernstein therefore gives
$$
\boxed{\mathfrak c.}
$$
No choice of an arbitrary infinite support is involved; it is the finite support that makes a <countable> union of finite-dimensional parameter spaces enough.
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