I have a problem to start my proof because I do not find a word $w$ where I can use the pumping lemma.
Task:
Be $\sum { =\left\{ a,b,c \right\} } $ and $S=\left\{ bx{ c }^{ m }|x\in { \left\{ a,b \right\} }^{ * }\wedge m\in N\wedge { |bx| }_{ a }-{ |bx| }_{ b }>m \right\} $
Proof with the the pumping lemma, that S is no regular language.
What I know is, that the word has to start we one $b$ following from n times $a$ or $b$ but $w$ needs to have one more $a$ than $b$ in it and $w$ also needs to have one more $b$ than $c$ in it to fulfill the condition. So it has to be something like this $W=b{a}^{2m+2}b^mc^m$? This language is very difficult for me, hope you see more than I.
P.S. Please no complete solutions of the full task.