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 $d(y_{q(n)-1},y_{p(n)})< \epsilon$.
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1So you're saying that for any $\epsilon$ there is N sufficiently large such that $|y_N-y_n|\le |y_N-y_{N+1}|+|y_{N+1}-y_{N+2}|+...+|y_{n-1}-y_n|\lt \epsilon$ for any $n\ge N$ but the sequence $(y_n)$ is not Cauchy? – Divide1918 Jun 29 '20 at 08:47
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What is the question? – Sahiba Arora Jun 29 '20 at 09:33
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Whether it is possible find such $\epsilon$, with the given hypothesis. – Earth Jun 29 '20 at 17:10