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A Pseudo-Cauchy sequence is: A sequence $(a_n)$ if $\forall$ $\epsilon > 0$, $\exists \ N \in \mathbb N$ such that $|a_{n+1} - a_n| \leq \epsilon\;\; \forall n \ge N$.

I know Pseudo-Cauchy sequence does not always converge. My question is what are the conditions for a Pseudo-Cauchy sequence to converge? For example, let a(n)=F(n+1)/F(n) where Fn is nth fibonacci number. In this example, a(n) is both a Cauchy sequence and pseudo-Cauchy sequence.

311411
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osdinuto
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  • https://math.stackexchange.com/questions/1535348, https://math.stackexchange.com/questions/1237655 – 311411 Mar 27 '22 at 19:26
  • @311411 I checked those pages before but neither of them answers my question. I am asking for conditions that satisfies pseudo-cauchy is cauchy. – osdinuto Mar 27 '22 at 19:29
  • We have a Latex-like typesetting system for mathematical expressions, called MathJax. Information is here: https://math.meta.stackexchange.com/a/10164 – 311411 Mar 27 '22 at 19:33
  • In the spirit of your question here is another that can offer you some insight into yours: Suppose that $E\subset \mathbb R$ is a bounded set. Find necessary and sufficient conditions on $E$ so that all pseudo-Cauchy sequences of points in $E$ are convergent. – B. S. Thomson Mar 27 '22 at 21:10

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