Subset of Countably Infinite Set is Countable

Theorem
Every subset of a countable set is either finite or countable.

Proof
Let $$S = \left\{{a_0, a_1, a_2, \ldots}\right\}$$ be countable.

Let $$T \subseteq S = \left\{{a_{n_0}, a_{n_1}, a_{n_2}, \ldots}\right\}$$, where $$a_{n_0}, a_{n_1}, a_{n_2}, \ldots$$ are the elements of $$S$$ also in $$T$$.

If the set of numbers $$\left\{{n_0, n_1, n_2}\right\}$$ has a largest number, then $$T$$ is finite.

Otherwise, consider the bijection $$i \leftrightarrow n_i$$.

This leads to the bijection $$i \leftrightarrow a_{n_i}$$

This latter bijection is the required one-to-one correspondence between the elements of $$T$$ and those of $$\mathbb{N}$$, showing that $$T$$ is indeed countable.