# Cauchy sequence

• Dec 17th 2009, 03:12 PM
platinumpimp68plus1
Cauchy sequence
Does anyone have a concrete example of a Cauchy sequence that is not convergent?
• Dec 17th 2009, 04:06 PM
Krizalid
i remember i studied once these, and every Cauchy sequence is convergent.

is there any counterexample, because i'm about to die.
• Dec 17th 2009, 04:12 PM
platinumpimp68plus1
Quote:

Originally Posted by Krizalid
i remember i studied once these, and every Cauchy sequence is convergent.

is there any counterexample, because i'm about to die.

Every convergent series is Cauchy... but in general, the converse isn't true. Only in R.
• Dec 17th 2009, 04:23 PM
Krizalid
oh, i'll have to review my notes.
• Dec 17th 2009, 04:33 PM
Jose27
Quote:

Originally Posted by platinumpimp68plus1
Does anyone have a concrete example of a Cauchy sequence that is not convergent?

It all depends on which spaces you're working. For example in working with $\displaystyle \mathbb{R}$ we have that $\displaystyle x_n=\left( 1+\frac{1}{n} \right) ^n$ is convergent, but if you're in $\displaystyle \mathbb{Q}$ this sequence is Cauchy but not convergent.