I have a vey specific question regarding the pumping lemma in the context of regular languages. The theorem states that if $L$ is a regular language, then there exists a constant $n$ such that for every string $w$ in L, with $|w|\geq n$, $w$ can be broken into $w=xyz$ with the following properties:
For all $k\geq 0$, $xy^kz \in L$
My question concerns the third and first properties. In the third property, does $k=0$ not imply that $|y|=0$, therefore contradicting the first property? As far as I can tell, when $k=0 \rightarrow xy^kz=xz$ and, consequently, $y=\epsilon$. What am I missing?