cardinal of the union of an increasing sequence

Question

Let $A_n\subset \mathbb{R}$ for all $n$. If $\vert A_n\vert=C$ for all $n$ where $C$ is the cardinality of the real numbers and $A_n\subset A_{n+1}$. then Is it true that

$\vert \bigcup_n A_n\vert=C$?

I appreciate any reference or suggestion .

Answer 1

$$\bigcup_{n=0}^\infty A_n\subseteq\Bbb R\implies |\bigcup_n A_n|\le C,$$ but $$A_1\subseteq\bigcup_nA_n\implies |\bigcup_nA_n|\ge C.$$