Let $\Sigma$ be an alphabet and let $L$ be a language on $\Sigma$. If it is known that all the equivalence classes of $R_L$ are finite, is $L$ regular? If definitely yes, prove. If definitely no, prove. If it cannot be determined give an example to each.
I can immediately say that if $L$ is finite, its equivalence classes of $R_L$ are finite, and it's regular. So I assume that $L$ is infinite. From Nerode theorem I know that $L$ is regular if and only if the number of equivalence classes is finite but it is known that all the equivalence classes are finite, so if it was regular it would have been finite, a contradiction, therefore if $L$ is infinite, and all the equivalence classes of $R_L$ are finite, then $L$ isn't regular.
So I know I need to find an example for both cases. For a regular, any finite language will suffice. But I'm having a problem finding an infinite language with finite equivalence classes