Pumping Lemma. Why is there a word w in L for infinite languages with n≤|w|≤2n
Suppose that $L$ is an infinite language which satisfies the pumping lemma with constant $n$. Let $w \in L$ be a word of minimal length in $L \cap \Sigma^{\geq n}$ (i.e., among the words in $L$ of length at least $n$, choose a word of minimal length). Since $L$ satisfies the pumping lemma, we can write $w = xyz$ in such a way that $|xy| \leq n$, $y \neq \epsilon$, and $xy^iz \in L$ for all $i \geq 0$. In particular, $xz \in L$. Since $|xz| &lt; |xyz|$, we must have $|xz| &lt; n$, and so $|w| = |xz| + |y| &lt; n + |xy| \leq 2n$.