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I am thinking yes, because a Cauchy sequence converges, so we can use the limit law for products twice, declare the cube of the sequence convergent, implying the cube is Cauchy. Is this correct? Are there general principles I am not aware of that would apply in wider circumstances?

NS248
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  • I think you might want to be more specific about where the Cauchy sequence lives but if you view the Cauchy sequence in the real or complex numbers then your reasoning is correct. A Cauchy sequence in the complex numbers converges by completeness and the cube of a convergent sequence converges, and convergent sequences are Caughy. – Seth Mar 15 '14 at 02:49
  • Thanks. I edited the post. I forgot about the wide application of Cauchy sequences. I would be happy to check your response as a correct answer if you decide to post it as one to make that possible. – NS248 Mar 15 '14 at 03:03
  • Ok, I made it an answer. Thanks! – Seth Mar 15 '14 at 03:15
  • No sequence is Cauchy (Cauchy was a mathematician, not a sequence), but some sequences are Cauchy sequences. – Michael Hardy Mar 15 '14 at 04:56
  • Can we not view Cauchy as the constant sequence, with every value being Cauchy? It is a very self referential sequence, and it is definitely Cauchy! – Seth Mar 15 '14 at 05:36

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Your reasoning is correct. A Cauchy sequence in the real numbers converges by completeness. The cube of a convergent sequence converges, and convergent sequences are Cauchy.

Seth
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