Some countability proofs

Theorem: The set of finite sequences of elements of a countable set is countable.

I like this result because it specializes to several other basic countability results: for example, it implies that countable unions, finite products, and the set of finite subsets of countable sets are countable. I know several proofs of this result and I am honestly curious which ones people prefer.

Proof 1

Map a finite sequence of positive integers to the positive integer . This map is a bijection by the uniqueness of binary expansion.

Proof 2

Map a finite sequence of non-negative integers to the positive integer . This map is a bijection by the uniqueness of prime factorization.

Proof 3

There are finitely many sequences of positive integers with for a fixed positive integer and a countable union of finite sets is countable. In the interest of being more explicit about the bijection here, one can order these sequences first by and then lexicographically.

Proof 4

If you believe that is countable, map a finite sequence of positive integers to the continued fraction . This map is a bijection by the uniqueness of continued fraction expansions (as long as they end in a positive integer greater than ).

Composing this map with the inverse of the map in Proof 1 gives an explicit enumeration of the rationals. I believe it was on the Putnam one year.

Question

Does anyone know any other genuinely different proofs?