Proof that a language is not regular as part of my homework assignment I have to proof that the following language is not regular using the pumping lemma:

$L = \{w\in \{a,b,c\}^* \; | \enspace z(a,w) = z(b,w) + z(c,w)\}$

The function $z(x,w)$ denotes the number of symbols $x$ in word $w$.

I started my proof by assuming that the language is regular. That means that there must exist an integer $m&gt;1$ so that $|w| \geq m$ is fulfilled.
 I have then chosen the following word which is part of L.

$w = a^{2m}b^mc^m$

After that I split my word in three parts ($x$, $y$ and $z$). Because $|xy| \leq m$ and $|y| &gt; 0$ must be true, at least one character $a$ has to be in $y$. 

I have read that I have to choose an $i$ such that the word $w_1 = xy^iz$ is not part of my language any more. I am not quite sure how I can proceed from above to finish my proof.

Could somebody please give some advice on this?
You've almost solved it already. You need to define which part of your word x, y, and z represent, and the obvious partition works well here. You know, by the pumping lemma, that you can repeat y any number of times, and the new word will still be in the language.
In this case, repeat y twice. Is the resulting word still a member of the language?