Let $A_1\subseteq A_2\subseteq A_3\subseteq...$ be a raising series of sets such that $\forall n\in \Bbb N \ |A_n|\lt |A_{n+1}|$. We mark $A$ as $A=\bigcup_{n\in\Bbb N}A_n$. Prove that $|A|<|A^{\Bbb N}|$
I've shown that $|A|\le|A^{\Bbb N}|$ since $f:A\rightarrow A^{\Bbb N}$ defined by $f(a)=(a,a,a,...)$ is an injection. I've considered for the sake of contradiction that $|A|=|A^{\Bbb N}|$ and tried to use Cantor's diagonal argument in order to get contradiction, but I got stuck. Thanks