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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7Yes. For example, take the sequence $\{e_i\}$ where $e_i$ is the $i$th standard unit vector. – 2012-12-06
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0And, I presume you meant "which contains no Cauchy subsequence". – 2012-12-06