Context-free Language - Pumping Lemma? im having some problems when it comes to Pumping Lemma of Context-free Language ... this is the Language : 

$$L = \{a^n b^n c^m | n\ge 0 , 0 \le m \le 2n \}$$

Here is my attemp to prove that the language is not context-free:

Assume $L$ is context-free. Let $p,r&gt;0$ be the pumping length given by the lemma.

Let $z=a^p b^p c^r$, then $z∈L$

Than according to the lemma, z can be written as z=uvwxy where the following properties hold:

$$|vx|≥1$$
$$|vwx|≤n$$
for every $i≥0$, $uv^iwx^iy∈L$

for the cases $i$ always had some problems ... there is one .. 

$vx = a^n b^n$ (no $c$'s )

i really have some problems understanding the PL for CF Language
The solution which I provide below is very similar to Example 2.37 in classical Sipser's book "Introduction to the theory of computation"


  Let $p,r&gt;0$ be the pumping length given by the lemma.


I think you have a little abuse of notation. Look carefully at definition. It states that if $L$ is $CFL$ then, there exists some constant $p$ (pumping length) that for every string $s \in L$ such that $|s| \ge p$, $s$ can be divided on five parts such that the following properites hold:


  $|vx|≥1$ $|vwx|≤n$ and for every $i≥0, uv^iwx^iy∈L$


So, let us take $s = a^nb^nc^{2n}$, where $n+n+2n &gt; p$. Then the conditions for lemma are hold, and it follows that $s = uvwxy$ with all those properties. From the first property we get that at least one of $v$ or $x$ is non-empty string. Now, we should consider several cases:

1) At least one of $v$ or $x$ contains diffent letters of the alphabet (for example, $v$ = $zabt$, $|z|, |t| \ge 0$). Then $uv^2wx^2y$ doesn't has the form $a^nb^nc^m$ (problems with order, in my example $v^2 = zabtzabt$), so we get a contradiction.

2) Either $v$ or $x$ contains only one letter from alphabet. Here we get such cases:

2.1) $v$ and $x$ don't contain the letter $c$. By lemma the string $s_1 = uv^0wx^0y \in L$. But $s_1$ has less than $n$ letters $a$ or $b$, while it contains $2n$ letters $c$. So, the property $m \le 2n$ doesn't hold in this case and we get a contradiction.

2.2) $v$ or $x$ contain the letter $c$, so either $a$ or $b$ doesn't belong to $v$ and $x$. Then string $s_1 = uv^2wx^2y$ contains at least $2n + 1$ $c$'s, and only $n$ $a$'s or $b$'s. Again, we get a contradiction.