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I have seen the statements "Every convergent sequence is a cauchy sequence" & "Every Cauchy sequence in R is convergent" . My question is that whether these statements means that a sequence is convergent if and only if it is convergent? The another statement for second statement is " R is complete metric space" . So what should i conclude ?

Kavita
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  • Note that in $\mathbb Q$, every convergent sequence is Cauchy, but not every Cuachy sequence converges. - (Albeit all this has little to do with Hausdorff) – Hagen von Eitzen Sep 25 '15 at 16:01
  • Both directions means iff. And that all Cauchy sequences converge is a definition of completeness, right. And what has all this to do with the Hausdorff (not droff!) property? – Sebastian Bechtel Sep 25 '15 at 16:01
  • Can you please make the title a relevant reflection of your question?? – rschwieb Sep 25 '15 at 16:02
  • I am.extremly sorry for the wrong title. Thanks for the answer. – Kavita Sep 25 '15 at 16:05

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One can easily prove that if $\langle x_n:n\in\Bbb N\rangle$ is a convergent sequence in a metric space, then it is a Cauchy sequence; that result is true for all metric spaces. It is not true in all metric spaces that every Cauchy sequence is convergent: that is the defining property of complete metric spaces, and not all metric spaces are complete. $\Bbb R$ with the usual metric, however, is complete, so it’s true that every Cauchy sequence in $\Bbb R$ with the usual metric is convergent.

Brian M. Scott
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