Show that every definite language is accepted by a finite automaton
This is true as a consequence of the Myhill Nerode theorem. For any fixed $k\gt 0$, the number of possible combinations of the last $k$ symbols of any string $w\in L$ is $|\Sigma|^k$, where $\Sigma$ is the alphabet.

Any two strings that have the last $k$ symbols identical will always be both accepted or both rejected by $L$, for any string appended to them, i.e., no string in $\Sigma^*$ will distinguish such strings. Thus, such strings lie in the same equivalence class of $L$.

Strings with differences in their last $k$ symbols will either lie in different equivalence classes or will cause the classes demarcated in the previous step to collapse together into a single class.

Thus, the equivalence classes of $L$ are determined by the last $k$ symbols of $L$.

As the possible combinations of the last $k$ symbols is finite, the number of equivalence classes is finite. Thus, by the Myhill Nerode theorem, the language $L$ is regular. Hence, there exists a DFA that accepts $L$.