Is the language $L=\{a^nb^mc^k | k \ge max (m,n) \}$ CF I'm trying to prove that the language $L=\{a^nb^mc^k | k ≥ max (m,n) \}$ is not CF. I consider the word $\alpha = a^pb^pc^p = uvxyz$ and i break the proof to the following cases :

1) if $vxy = a^l, b^l$ or $a^kb^l$ for some $k,l \le p$ i pump the word up, so 
$a^l+s, b^l+s, s &gt; 0 : l+s&lt;p =&gt; \alpha \notin L$

2) if $vxy$ consists any $c$ then it can't consist any $a$s so if we pump down then $c^u, u = p - l, l &gt; 0 =&gt; N_{c} &lt; N_{a} =&gt; \alpha \notin L$.

But i keep thinking relying on my intuition that L is CF, because L build on the rule : if we print a then we print c, if we print b we print c, or we just print c. Looks like CF.
My question is is my proof wrong or my intuition?
Your intuitive argument is wrong. If you print a $c$ for every $a$ or $b$ you see, then the language will have $k\ge m+n$ rather than $k\ge \max\{m,n\}$.