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The definition of Cauchy sequence is: for any $ε > 0$, there exists a natural number $N$ such that if $m, n ≥ N$, then $|a_m − a_n| < ε$.

What if we changed the definition to: for any $k ≥ 1$, there exists a natural number $N$ such that $|a_{n+k} - a_n| < ε$ for any $n ≥ N$.

What is the difference between these 2 definitions. Will the second one be true as well?

Thank you!

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  • $\begingroup$ You seem to have lost "for any $ε > 0$" in your second definition but then use $ε$. Condition order matters: "for any $ε > 0$ and for any $k ≥ 1$, there exists a natural number $N$ such that ..." is a weaker condition than "for any $ε > 0$, there exists a natural number $N$ such that for any $k ≥ 1$ ..." $\endgroup$
    – Henry
    Oct 4 at 21:40

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$a_n=\ln n$ satisfies the second criterion but not the first. The first one implies the second by triangle inequality. [$|a_{n+k}-a_n|\leq |a_{n+k}-a_{n+k-1}|+|a_{n+k-1}-a_{n+k-2}|+\cdots |a_{n+1}-a_n|$. Make each term less than $\epsilon /k$].

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  • $\begingroup$ thank you! I can see how the first one implies the second now. But I don't see how $a_n = ln(n)$ satisfies the second criterion. $\endgroup$
    – All is number
    Oct 4 at 13:07
  • $\begingroup$ Use: $\ln a-\ln b=\ln (a/b)$. Also, $\ln (1+x) \le x$ . $\endgroup$
    – geetha290krm
    Oct 4 at 13:16

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