Please help me understand the following
$L = \{ a | a ∈ \{0, 1\}^∗, |a| = k ≥ 4, a = a_1a_2...a_{k−1}a_k, ∃i ∈ N, 1 ≤ i < k : a_i = a_{i+1} \}$
To prove: The language $L$ has regular pumping property
My argumentation:
Let $p = 4 $. Let $z$ in $L$, with $z$ $\geq $ $p$. Set $u$ to $\epsilon$. Than $x$ is the first letter, and $w$ the rest.
$x$ $\neq$ $\epsilon$
For all $i$ $\geq $0$: ux^iw$ is in $L$
But if i set i to 0 the word is not in the language? And therefor does not have the regular pumping property? What am i missing? Example $0010$, after pumping $4$ $\geq $ $010$?