$\aleph_0\times 2^{\aleph_0} = 2^{\aleph_0}$ requires the Axiom of Choice? In a model solution it is stated that the cardinality of a set which is the countable union of sets of cardinality $2^{\aleph_0}$ is $\aleph_0\times 2^{\aleph_0}$, and, using the Axiom of Choice, $\aleph_0\times 2^{\aleph_0} = 2^{\aleph_0}$.
But, isn't the Cantor/Shroeder-Bernstein Theorem enough? I mean, given this:
$2^{\aleph_0}\le \aleph_0\times 2^{\aleph_0}\le 2^{\aleph_0}\times 2^{\aleph_0} = 2^{\aleph_0+\aleph_0} = 2^{\aleph_0}$
EDIT: so, the set in question is the set of all real-valued sequences $s = (s_0,s_1,\dots)$ such that for all but finitely many $n$, $s_n = 0$. And so I see  that we have to get a countable union of sets of size $|\Bbb{R}^n|=2^{\aleph_0}$ for each $n$. Is there an (hand-wavy should be fine) way to set up the bijection needed with AC?
 A: Your solution is fine, even without the Axiom of Choice.
Choice is needed in order to move from $|A|\cdot|B|=|\bigcup_{a\in A}B_a|$, where $B_a$ is some set of cardinality $B$. Namely, infinite unions are not necessarily the same as a product, not as freely as we take them when we assume AC.
If, however, $B_a=\{a\}\times B$, for example, then that's fine, as the union is exactly $A\times B$, which is by definition the set whose cardinality is $|A|\cdot|B|$.

So $\aleph_0\cdot2^{\aleph_0}$ is the cardinality of $\Bbb{N\times R}$. As you identify correctly, that is a subset of $\Bbb{R\times R}$, which without choice, has the same cardinality as $\Bbb R$ itself. And finally, Cantor–Bernstein is also choice-free, so we are done.
It's not true, however, that a countable union of sets of size $2^{\aleph_0}$ each is also of size $2^{\aleph_0}$. For example, if $A$ is a set which is a countable union of countable sets $A_n$, but it cannot be linearly ordered (e.g., it is a union of finite sets), then taking $R_n=\Bbb R\times\{n\}\cup A_n$. Since each $A_n$ is countable (or finite), we get that $R_n$ has size $2^{\aleph_0}$.
But $\bigcup R_n=\Bbb{R\times N}\cup A$, so it cannot be linearly ordered and thus has cardinality strictly bigger than $2^{\aleph_0}$.
However, if you can uniformly biject $R_n$ with $\Bbb R$, then you can turn this into a bijection into $\Bbb{R\times N}$, and the cardinality is again $2^{\aleph_0}$. This is indeed the case where $R_n$ is something like $\Bbb R^n$.
Perhaps easier, note that $|\Bbb{R^N}|=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}$. So any set that can be mapped into $\Bbb{R^N}$ has cardinality of at most $2^{\aleph_0}$. And indeed, any $\Bbb R^n$ can, as well as the eventually $0$ sequences.
A: You're right, the axiom of choice is not needed to show that $\aleph_0\cdot 2^{\aleph_0}=2^{\aleph_0}$.  Note though that the axiom of choice may still be needed in the solution you refer to, specifically to say that since your set is a countable union of sets of cardinality $2^{\aleph_0}$, its cardinality is (at most) $\aleph_0\cdot 2^{\aleph_0}$.  In general this requires choice, since you need to simultaneously pick bijections which witness that each of the countably many sets has cardinality $2^{\aleph_0}$.  (In many cases, though, there is an easy way to construct such a family of bijections without the axiom of choice.)
