For a language $L$ over an alphabet $\Sigma$, we say that two words $v,w \in \Sigma^*$ are equivalent, denoted $v\sim w$, if for every word $z \in \Sigma^*$, $vz \in L$ iff $wz \in L$. We define $[w]_L$ to be the equivalence class of $w$ under this relation. The index of $L$ is the number of equivalence class.

How can I prove that every regular language has a finite index?
can I use the Myhill-Nerode theorem?

I tried to use the fact that if $\delta^*(v) = \delta^*(w)$ then $v \sim w$.