$($i$)$ In a DFA, we require the set of states to be a finite set. Let$$s define a IA $($Infinite Automaton$)$ in the same way as a DFA, except that the set of states may be any set $($not necessarily finite$).$ Prove, by formal proof, that every language is recognized by a IA$.$ Be sure to completely and unambiguously describe your DIA, for example as a $5-$tuple.

$($ii$)$ Suppose L is an infinite regular language. Prove that there are two infinite regular languages L$1,$ L$2$ with L$1$ union L$2=$ L and L$1$ intersection L$2=.$