= Solution
The common amplitude $a>0$ is uniquely determined by any entry's <absolute value>. After choosing it, each entry independently has sign $+1$ or $-1$. Thus the correspondence
$$
(a,(\varepsilon_n)_{n<\omega})\longmapsto(a\varepsilon_n)_{n<\omega}
$$
is a <bijection> from $(0,\infty)\times\{-1,1\}^{\mathbb N}$ onto the <set>. The positive reals have cardinal $\mathfrak c$, and binary sign <sequences> have cardinal $2^{\aleph_0}=\mathfrak c$. Hence
$$
\boxed{\mathfrak c\cdot\mathfrak c=\mathfrak c.}
$$
Here “independently” refers to unconstrained deterministic choices of coordinates, not probabilistic independence; no probability distribution is being specified.
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