# Proof that $\{a^ib^jc^k\mid i,j,k\in\mathbb{N}, i<k<j\}$ is not context-free using the Pumping Lemma

$$L=\{a^ib^jc^k \;| \;i, j, k \in \mathbb{N} \; \text{and} \; i

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

1. only $$a$$'s
2. only $$b$$'s
3. only $$c$$'s
4. $$a$$'s and $$b$$'s
5. $$b$$'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!

Not you don't have to find a pumping constant. To the contrary, you have to show no such constant can exist. So, the general argument is usually like "if I assume $$N$$ is the pumping constant, I can use this word $$x\in L$$, longer than $$N$$, and whatever I try, we cannot pump it and stay in $$L$$."
Usually one choses a string that is "just" inside the language, in this case $$a^Nb^{N+1}c^{N+2}$$. Now check your cases. What if we pump $$a$$'s and or $$b$$'s, but no $$c$$'s etcetera.