Using Pumping Lemma to prove language $L = \{(01)^m 2^m \mid m \ge0\}$ is not regular
Hint: You need to consider what all the decompositions $w=xyz$ look like, so what are all the possible things $x$, $y$ and $z$ can be given that $xyz=(01)^p2^p$. Then you pump each one and see whether you get a contradiction, which will be a word not in your language. Each case needs to lead to a contradiction, which would then be a contradiction of the pumping lemma. Voila! The language would not be regular.

Of course, you need to work through the details and consider all the possible splittings.