Pumping lemma for regular languages. Proof
I assume that you want to show that the language $L$ does satisfy the Pumping Lemma, as it indeed is regular. In words your language consists of all strings that have length at least 4 and contain two consecutive symbols that are the same. The position of a pair of equal letters letters is indicated here with $i$. (Note $i$ in general is not unique.)
So we need to explain that all strings in the language (above a certain length) can be pumped within the language.
You must realize that your division into $uxw$ can actually depend on the string $a$ itself.
So if $a$ starts with a repeated letter, e.g., $a = 00\dots$, we can pump the third letter. Otherwise if the repeated pair is assumed to be not in the first position, we can then pump the first symbol and we keep the repetition in the string while pumping.