Is there a bounded sequence in $l^2= \{a \in \mathbb{R}^{\mathbb{N}} |\sum_{k=0}^{\infty} |a_k|^2 < \infty \}$ which contains no Cauchy subsequence?
Can one find a bounded sequence in $l^2$ which contains no Cauchy sequence?
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calculus
real-analysis
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0And, I presume you meant "which contains no Cauchy subsequence". – 2012-12-06
1 Answers
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Yes. Let $e_n$ be the vector in $\ell_2$ whose $i$'th coordinate is $1$ if $i=n$ and $0$ otherwise. Note $\Vert e_n\Vert=1$ for each $n$. $(e_n)$ is bounded.
It's not hard to verify that $\Vert e_n-e_m\Vert=\sqrt2$ whenever $n\ne m$. From this it follows that no subsequence of $(e_n)$ is Cauchy.
(More generally, such a sequence can be found in any infinite dimensional normed space, using Riesz' Lemma. See this).