# Pumping lemma for context-free languages: Importance of length restriction

(from 'An Introduction to Formal Languages and Automata' by Peter Linz)

What I do not understand, is why we have done our best to make sure that the condition (8.2) holds. Why is this restriction important/useful?

The restriction is definitely useful. It provides you with an additional guarantee on the structure of the decomposition of $$w$$, which you can exploit in your proofs.
For example the language $$L = \{a^n b^k c^n d^k \mid n, k \ge 0\}$$ is not context-free and a possible proof using the pumping lemma is as follows: suppose that $$L$$ is context free, and let $$m$$ be its pumping length. Consider the word $$w=a^m b^m a^m b^m$$. The pumping lemma ensures that there is a decomposition of $$w = uvxyz$$ with $$|vxy| \le m$$, and this implies that $$vxy$$ cannot contain both a $$a$$ and a $$c$$, nor both a $$b$$ and a $$d$$. As a consequence $$uv^2xy^2z \not\in L$$, which provides the sought contradiction.
Notice that the above proof would not work without using $$|vxy| \le m$$, since we could pick $$u=a^{m-1}$$, $$v=a$$, $$x=b^m$$, $$y=c$$, and $$z=c^{m-1}d^k$$, which satisfies $$uv^i x y^i z \in L$$ for all $$i \in \mathbb{N}$$.