The following statement and proof illustrate a formal structure that can be used to make an argument based on the pumping lemma for context-free languages.
Let A = {an bn cn | n ≥ 0}. Then A is not a context-free language.
Let k by an arbitrary non-negative integer. Set z = ak bk ck. Then |z| ≥ k.
Let u, v, w, x, and y be any strings for which z = uvwxy, vx ≠ ε, and |vwx| ≤ k. The structure of v, w, and x breaks down into the following cases:
There are no more cases than cases 1 and 2 since |vwx| ≤ k. Thus, for any decomposition of z into uvwxy with the properties listed above, there exists an i ≥ 0 (specifically, i = 2) such that uviwxiy ∉ A. Therefore, by the Pumping Lemma, A is not context-free.
To make a PL proof persuasive and readable, your best bet is to follow the rigid structure as above.