 Solution

For an infinite cardinal $\kappa$, multiplication by a nonzero countable cardinal does not change it. Since $|\mathbb{R}|=\mathfrak{c}\ge {\aleph }_0$,There is also an immediate lower bound from $r\mapsto (r,0)$; the standard product identity supplies the matching upper bound. The discreteness of the second coordinate does not create a new cardinal beyond the continuum.