# a^*b^*c^* – {a^n b^n c^n | n ≥ 0} is not regular using pumping lemma

$$L=a^*b^*c^* \setminus \{a^n b^n c^n \mid n \geq 0\}$$ can be proved as context-free by partitioning it as $$L = \{a^nb^mc^* \mid n \neq m\} \cup \{a^*b^nc^m \mid n \neq m\}$$ and further dividing each $$\neq$$ into smaller and larger. You will have four sets. You can give CFG’s for each. Then since CFGs are closed under union, you have your proof.

Now how can I go around proving that $$L$$ is not regular? If you prove that the each of these four sets are not regular, you still can not prove that the union is not regular, can you?