Prove $\bigcup _{i=1}^\infty A_i$ is not regular. We know $A_i$ is regular, but how can prove the infinite union is not regular. I think a counter example would work, but I can't think of any. Conceptually, I understand the infinite union can't be regular because you can't map infinity with a finite state machine.
A counter example of the infinite intersection not being regular is $0^*1^*-0^i1^i$.
In an attempt to answer my own question: {0}, {01}, {0011}, {000111}... are all regular, but the union of all these languages $\{0^i1^i\}$ is clearly not regular. Does that work?
There are only countable DFA's but non countable number of languages, thus there is some countable infinite language which is not regular. Write that language as the union of the singletons of the words in the language