My goal is to show, that a given language is not a regular one by using the Properties of Regular Languages.
The language is $ A \triangleq\left\{w \in \Sigma^{*} \mid |\left.w\right|_{b} \neq|w|_{c} \wedge|w|>0\right\} $ with $ \Sigma \triangleq\{a, b, c\} $
My idea is to use this clearly non regular language (which is given in the task, so I can/should use it) $ \left\{b^{n} c^{n} | n \in \mathbb{N}\right\} $ to prove $A$ is also no regular language.
Could the following work?
$(A\cap L(b^*c^*)) \cdot \mathrm{L}(\boldsymbol{\epsilon}) = \left\{b^{n} c^{n} | n \in \mathbb{N}\right\}$