Please forgive me if this is a very basic question. I have never studied cardinals, I just need this result for a problem in Linear Algebra.
My question is: if $V_1, V_2, ..., V_n$ are infinite sets of the same cardinality, does $V_1 \cup V_2 \cup \cdots \cup V_n$ still have that same cardinality?
I can imagine a proof if the sets are countable: we can list the $V_i's$ in $n$ rows, and construct a "snake". However, I don't see how to generalize this to uncountable sets.