I came across this: https://math.stackexchange.com/a/407588/820472 interesting claim regarding the Cauchy property of sequences in metric spaces, which applies the Big O. I would like to read more about Big O and its applications in the proofs for analysis and metric spaces, but unfortunately I don't know any sources for this. Thus I'm asking for some written reference for this claim, and if possible an explanation for the proof of the claim. Currently my biggest stumbling block is showing that a such function $f$ must exists.
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$\begingroup$ en.wikipedia.org/wiki/Big_O_notation $\endgroup$– AlanJul 23, 2021 at 5:53
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$\begingroup$ @Alan That article doesn't contain discussion about the Cauchy sequence. $\endgroup$– Epsilon AwayJul 23, 2021 at 6:06
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1$\begingroup$ I dispute what it says in the linked answer. While the condition given is sufficient, I don't believe it is necessary to achieve Cauchiness. That is, there are Cauchy sequences that fail to satisfy this condition. For example $x_n = \frac{1}{\sqrt{n}}$. $\endgroup$– Theo BenditJul 23, 2021 at 6:29
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