Let's define the following operations:
$odd(string) = $ odd characters of $string$, $even(string) = $ even characters of $string$
Now say we have some language $L$, we will then define the following languages:
$odd(L) = \{odd(w): w \in L\}$, $even(L) = \{even(w): w \in L\}$
Need to prove that there exists a non-regular language, $L_N$, such that $odd(L_N)$ and $even(L_N)$ are both regular languages.
I thought of a few examples that might produce obvious regular languages with $odd$ and $even$, one of which was the language:
$L_1 = \{w \in \{a,b\}^*: |a|_w = |b|_w\}$, where $|x|_w =$ # of $x$'s in the string $w$
I know how to prove that this is non-regular, and upon inspection of a lot of values of $odd(L_1)$ and $even(L_1)$, it appears both produce the language $\{a,b\}^*$ which is obviously a regular language. Don't really know how I could prove that equality though.