$\ \sum= \{a,b\} $ Show that: $ L:= \{a^nwb^n: m,n \in \mathbb N, m\geqslant n, w\in\sum^m\} $ is not regular.
I'm trying to proof this with the Pumping Lemma, but I'm kind of confused because of the middle part "w".
I would pick the symbols $\ a^na^mb^mb^n $ as a starting point and show that this does breach at least one of the three properties of the pumping lemma. At this point I don't know if this is the right word to pick & how I do continue from here.
I appreciate any kind of help. Thank you!
You’re having trouble because $L$ is regular. If by $\Bbb N$ you mean the set of non-negative integers, then $L=\Sigma^*$, because any word $w\in\Sigma^*$ can be written in the form $a^0wb^0$, where $|w|\ge 0$. If by $\Bbb N$ you mean $\Bbb Z^+$, the set of positive integers, then
$$L=\left\{awb\in\Sigma^*:|w|\ge 1\right\}\;,$$
which corresponds to the regular expression $a(a\lor b)(a\lor b)^*b$, or if you prefer, $a(a\lor b)^+b$. (You might be more familiar with a different notation: $a+b$ and $a\mid b$ are also used where I have $a\lor b$.) You can easily design a regular grammar that produces this language.