Is $L=\{a^ib^jc^k : i\leq j\leq k\}$ context-free? I have an exam and in preparation I found this language. We are supposed to determine where in Chomsky hierarchy it stands. The language is $L=\{a^ib^jc^k : i\leq j\leq k\}$. I can easily build a linear turing machine, that would delete a, b and c for each a and then b and c for each b and if the only thing left, will be c, then the word belongs to the language. So I know the language is context-sensitive. What I can't figure out is if the language is context-free. 

I tried to build a pushdown automata but I couldn't find any. I also tried to use the pumping lemma but wasn't able to find a good word to prove it's not context-free. 
Could you help me?
A word like $a^nb^nc^n$ should work. Remember that the pumping lemma also allows you to pump down, i.e., delete the pumping segments. 

Also, each splitting $z=uvxyz$ can have its own "contradiction" to show it does not adhere to the pumping lemma. You can use different arguments, so different $i$ for each splitting. That is very natural here as $a$ has the minimal number of occurrences, and contradiction is reached by adding occurrences, whereas for $c$ deletion (choosing $i=0$) will get you into trouble.