Checking whether union of two languages is regular How to check if 

$L = \{c^ka^nb^n \mid k&gt;0 \wedge n\geqslant0\} \cup \{a, b\}^*$ 

is regular ,where 

$L_1 = \{c^ka^nb^n \mid k &gt; 0 \wedge n\geqslant0\}$ is clearly not regular and $L_2 = \{a, b\}^*$ is... ?
If $L$ were regular than so would the following language be:
$$L \cap ca^*b^* = \{ ca^nb^n \mid n \geq 0 \}.$$
You can show that the latter language is not regular in various ways.