$$

L={a^ib^jc^k ;| ;i, j, k in N ; and ; i <k<j}

$$

I need to show that this language is not context-free with the help of the Pumping Lemma.

My first intuition is, that there exist 5 different cases, i.e. the middle part, let’s call it vwx, consists of

- only a’s
- only b’s
- only c’s
- a’s and b’s
- b’s and c’s

and I need to find a pumping constant, which excludes the new word from the above defined language. However, I am having a hard time how show that formally and precisely. Any hints are highly appreciated!