Is infinite union of RE a RE?

Question

Is an infinite set of RE languages create a language that is also RE?

I think it's true, and my first intuition is to try induction to prove this statement.

Am I on the right way?

Thanks!

Answer 1

The statement is not true; any language $L$ can be written as the (countably infinite) union of singleton languages which each just contain one word of $L$.

Also, even if the statement was true, one could not prove it using (regular) induction. This is because the finite union of RE languages is indeed RE. My go-to example here is that one can show using induction that every natural number is finite, but there exists no infinite natural number.