The Cartesian product of a finite number of countable sets (each with cardinality > 1) is countable. However, the Cartesian product of a countably infinite number of countable sets (each with cardinality > 1) is uncountable. For example, see here and here.
Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets (each with cardinality > 1) and let $$S=E_1\times\cdots\times E_n\times\cdots $$
So $S$ is uncountable. Now let $$T_n=E_1 \times E_2 \times \cdots E_n $$
So loosely speaking, as $n$ approaches infinity, $T_n$ becomes uncountable. A friend of mine has trouble grasping this, as in calculus, for example, the limit of an infinite series can be approximated by finite partial sums. My explanation is that Cantor's diagonal argument used to prove the uncountability doesn't work for finite sets, and "countability" is not something defined using epsilon-delta.
1.Can anyone think of a better and more intuitive explanation? Or an explanation that is more to the point?
2.What are some other (preferably simple) examples, where "the limit in infinity is not consistent with the finite processes that lead up to it"?