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I am having difficulty in grasping Cauchy sequence. They say it is a sequence $\{{x^{(k)}}\}$ such that

$$\lim_{m\to\infty} d(x^{(i)},x^{(m)}) = 0.$$

I didn't get what sort of sequence is it. Just explain me basically.

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2 Answers 2

In fact, that definition does not make sense as the expression depends on $i$ (taken literally, it would imply the sequence is constant!). Rather, a Cauchy sequence $\{x^{(k)}\}_{k\in \mathbb N}$ (in a metric space) is a sequence such that for any $\epsilon>0$ there is an $N\in\mathbb N$ such that $i,k>N$ implies $d(x^{(i)}, x^{(k)})<\epsilon$. If the metric space is complete then Cauchy sequences are precisely the converging sequences.

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To show a sequence converges the usual way you must first guess a limit and prove that the sequence eventually gets within any positive distance of this guess and stays there.

One might think this forces not only terms to get close to the limit but terms to get close to other terms. In fact it does in some sense.

The notion of Cauchy sequence (as defined above by Hagen) measures the ability of terms to get close to other terms and to stay close. The point here is that such sequences look like they should converge but in some cases they don't (for example we can consider working with rational number sequences and take a sequence of rational approximations to $\sqrt{2}\notin\mathbb{Q}$).

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