$4.(20$pt$)$ Prove that the following problems are decidable:

$($a$)(10$pt$)$ Given a regular language $\backslash ($ L $\backslash )$ over $\backslash (\backslash \{$a$,$ b$\backslash \}\backslash ),$ is it true that $\backslash ($ L $\backslash )$ contains infinitely many strings that have at least one $\backslash ($ a $\backslash )$ but finitely many strings that have at most one $\backslash ($ b $\backslash )?$

$($b$)(10$pt$)$ Given two regular languages, $\backslash ($ L$\_\{1\}\backslash )$ and $\backslash ($ L$\_\{2\}\backslash ),$ is it true that every string in $\backslash ($ L$\_\{1\}\backslash )$ is a prefix of a concatenation of strings from $\backslash ($ L$\_\{2\}\backslash )?$

You are allowed to use any of the following:

$-$ closure properties: union, concatenation, Kleene star, complement, intersection, difference

$-$ conversion algorithms between DFSM$,$ NDFSM$,$ regular expressions, and regular grammars $($see the last slide of Ch$\mathrm{.}7$: Conversions$)$

$-$ decision algorithms: membership, emptiness, finiteness, totality, equivalence, minimality.

Explain clearly which of the above closure property and algorithm you have used at each step. Any other construction or algorithm should be described in the assignment.