Let $(a_i)_{i \in I}$ and $(b_j)_{j \in J}$ be two families of cardinal numbers. Then,
(a) Is it true that $\sup (a_ib_j)_{i \in I,j\in J}=\sup (a_i)_{i \in I}\sup (b_j)_{j\in J}$?
what about the infimum?
(b) Also, can we say $\prod_{i \in I} a_i=\sup (a_i)_{i \in I}$ if at least on element of $(a_i)_{i \in I}$ is an infinite cardinal?
Note. The supremum of any set of cardinals is again a cardinal (see this and this), and we know both (a), (b) are true if $I$ and $J$ are finite sets.
It is easy to show that (a) holds, clearly, $a_ib_j\le (\sup_i a_i)(\sup_j b_j)$, so $\sup(a_ib_j)\le (\sup_i a_i)(\sup_j b_j)$. Similarly, $a_ib_j\le \sup(a_ib_j)$ proves the remaining inequality.
However, (b) is not true: for example, you can see that $$\prod_{n<\omega} \aleph_n = \aleph_{\omega}^{\aleph_0},$$ and this is $\aleph_{\omega+1}$ if GCH holds. However, $\sup_n\aleph_n=\aleph_\omega<\aleph_{\omega+1}$.