Proving union of two $\sigma$ algebras in $X$ is an algebra, then it is a $\sigma$-algebra.

Question

I tried two countable union of $M_1, M_2 \in 2^X$ such that $A\in M_1, B\in M_2$ where $ A=\bigcup_{k=1}^{\infty}A_k, B=\bigcup_{k=1}^{\infty}B_k $. But Im stuck on that I should prove

$[\bigcup_{k=1}^{\infty}A_k] \cup[\bigcup_{k=1}^{\infty}B_k]=\bigcup_{k=1}^{\infty}[A_k\cup B_k]?$

If $k=1,2,3, ...$