Your friend claims that any language can be recognized by a finite automaton. Here

is their argument:

$($i$)$ A language is a set of strings. Any language A can be rewritten as A $=$ S

w in A

$\{$w$\}.$

$($ii$)$ For any string w$,\{$w$\}$ is a regular language since we can use the regular

expression w to specify a language that only contains w$.$

$($iii$)$ By $($i$)$ and $($ii$),$ A can be represented as a union of regular languages.

$($iv$)$ The union of any two regular languages is also a regular language; that is$,$

regular languages are closed under union.

$($v$)$ By $($iii$)$ and $($iv$),$ A itself must be regular as it is a union of regular languages.

$($vi$)$ Since A is regular, there exists some finite automaton that recognizes it$.$

Therefore, any language has a finite automaton that recognizes it$.$

Which steps of this argument are valid and which are not? Explain your answers.