Indexed Union of Countably Infinite Sets

Prove that if $\displaystyle \Lambda$ is countably infinite and $\displaystyle B_{\lambda}$ is countably infinite for each $\displaystyle \lambda \in \Lambda $, then $\displaystyle \bigcup_{\lambda \in \Lambda}B_\lambda$ is countably infinite.

I was taught that in order to show that a set is countably infinite, we need to either find a proper subset with the same cardinality, or find a bijective function from the domain to the codomain. However, I'm not sure how to attack this proof; I don't know if I'm supposed to use the above method since I'm not just proving that one set is countably infinite.