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I need to prove that ${\{X_n\}}$ is not a Cauchy sequence. I understand that in order to prove this, I need to prove that $$$$$\exists\ \epsilon\gt0\ | \forall N \in \Bbb N,$ if, there is $n,m \geq N$, then $|X_n-X_m|\geq \epsilon$

So my question, what happens if $n=m$?, then $|X_n-X_m|=0$ and therefore $|X_n-X_m|<\epsilon$ because $\epsilon \gt 0$ and it would never hold, because for $\forall N$ there would exist an $n,m$ that satisfy this. So, do the $n,m$ need to be different?

Thanks for any help! (:

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3 Answers 3

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To assert that $(X_n)_{n\in\mathbb N}$ is not a Cauchy sequence means that$$(\exists\varepsilon>0)(\forall N\in\mathbb N)(\exists m,n\in\mathbb N):m,n\geqslant N\text{ and }\lvert X_m-X_n\rvert\geqslant\varepsilon.$$So, you don't need to worry about the $m=n$ possibility. Given $N\in\mathbb N$, you only need to prove that there is some $m\geqslant N$ and there is some $n\geqslant N$ such that $\lvert X_m-X_n\rvert\geqslant\varepsilon$. And, of course you will not pick $m=n$.

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We are not saying that $|X_n-X_m| \geq \epsilon$ whenever $n,m \geq N$. We are only saying that this holds for some $n,m \geq N$, Surely such $n$ and $m$ cannot be equal.

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A little knowledge of quantifiers may help you here. The negation of universal quantifier '$\lor$' (for all)is existential quantifier '$\exists$' (for some). Now just negate the Cauchy's criterian of existence of limits to get the one you needed.

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