A construction in a non Cauchy Sequence

Question

Let $(y_n)$ be a sequence which is not Cauchy. Let $\lim_{n\to\infty} d(y_n,y_{n+1})=0$ and $d(y_n,y_{n+1})$ is a decreasing sequence. Then there exists an $\epsilon>0$ such that for every $n\in \mathbb N$, there exists an odd integer $q(n)\in \mathbb N$ and an even integer $p(n) \in \mathbb N$ with $n<p(n)<q(n)$, $d(y_{p(n)},y_{q(n)})\ge \epsilon$ and $d(y_{q(n)-1},y_{p(n)})< \epsilon$.