How can I remember whether finite or countable cartesian product of countable set is countable

Question

I always forget this result

Is cartesian product of countable set countable under finite or countable cartesian products?

Is there a good way to remember this? Like a proof sketch where the proof would go wrong if we take the product of too many countable sets?

Answer 1

Suppose each of your countable sets actually is $\mathbb{N}$. Then a countable product of them would have elements of the form

$$(n_0,n_1,\dots)$$

i.e. an infinite sequence of natural numbers.

Any real number in $[0,1]$ has a decimal expansion of the form

$$0.d_0d_1\dots$$

i.e. an infinite sequence of digits in $\{0,\dots,9\}$.

Since $\{0,\dots,9\}\subseteq \mathbb{N}$, there are at least as many sequences of natural numbers as there are reals in $[0,1]$, i.e. uncountably many.