We know for $A_1, A_2, \cdots, A_n$, where each $A_i$ is countable, that $A_1 \times A_2 \times \cdots \times A_n$ is countable.
We also know that $\mathbb{N}$ is countable.
Let $N_i = \mathbb{N} \times \mathbb{N} \times \cdots \times \mathbb{N}$ ($i$ times).
I am wondering if $\bigcup_{i=1}^\infty N_i$ is countable?
This is bothering me because for any given $i$, we know that $N_i$ is countable, so we have a countable union of countable sets, therefore, $\bigcup_{i=1}^\infty N_i$ should be countable.
But then we have that as $i\to\infty$, $N_i$ is countable. This implies that an infinite cartesian product of countable sets is countable, which is not true. It is uncountable.
Where am I going wrong and which is correct?
Yes, $\bigcup_{i=1}^\infty N_i$ is countable.
It is the collection of all finite sequences of integers,
a union of countablely many countable sets.
It is disjoint from N$^{\aleph_0}$,
the set of all infinite sequences of integers.