Proof that $L=\{a^ncb^n| n \ in \mathbb{N}\}$ is not regular


Proof that $ L=\{a^ncb^n| n \in \mathbb{N}\}$ is not regular.

Here is my try, I would really appreciate if someone could tell me if this is a correct proof.

Lets assume L is regular. Then we know that L must meet the requirements of the pumping lemma. So let p the pumping number.

Let $ w=a^pcb^p$ . $ w$ is obviously of the length p and is in L. Therefore it should be possible to split w into three pieces xyz such that $ |y|>0,|xy|<=p,xy^iz$ is in L $ \forall i \in N$ . Because $ |xy|<=p$ $ y $ can only contain the symbol $ a$ (If $ y$ would contain a symbol different from a it would implicate that $ |xy|>p$ , which is not possible). Therefore $ y$ must be in the form $ y=a^{p-k},k<=0<p$ . So the word w equals $ w=a^ka^{p-k}cb^p$ , if we set i=2 we get $ a^ka^{2p-2k}=a^{2p-k},k<p$ and because $ a^{2p-k},k<p\neq b^p$ it follows that the pumped $ w$ is not in $ L$ . Which is a contradiction. Therefore L is not regular.

$ q.e.d$