Proving the set of finite languages is countable without using the union of countable sets
Let your finite alphabet be $\Sigma = \{a_1, \dots, a_\ell\}$ and let $\#$ be some character not in&nbsp;$\Sigma$. Let $L=\{w_1, \dots, w_n\}$ be a finite language over&nbsp;$\Sigma$. You can consider the string $\#w_1\#w_2\#\dots\#w_n$ to be a number in base&nbsp;$|\Sigma|+1$ by associating the symbols $a_1, \dots, a_\ell, \#$ with the base-$(|\Sigma|+1)$ digits $0, \dots, \ell-1, \ell$, respectively (starting the string with $\#$ ensures that the leading digit isn't zero). This gives a map from the set of finite languages over&nbsp;$\Sigma$ to a subset of the integers, so that set of languages is countable.