elementary set theory - What could be said about the cardinality of $bigcup_{iin I} A_i$ if $I$ and all the $A_i$ have cardinality $2^{aleph_0}$

The countable union of a countable set is countable. Does the same hold for sets with cardinality $|\mathbb R|$. More specifically, if $A_i$ are sets of the same cardinality as the real numbers, and $I$ is an index set also with cardinality $|\mathbb R|$, is $|\bigcup_{i\in I} A_i| = |\mathbb R|$?

Answer

This question can be reduced to the question "is it true there is a bijection between $\mathbb{R}$ and $\mathbb{R}^2$?" The answer is yes.