Define $\ell^1=\{x\colon\mathbb N\to\mathbb F: \|x\|_1~\mbox{is finite}\}$ where $\mathbb F$ is either $\mathbb R$ or $\mathbb C$. If $(x_n)$ is a Cauchy sequence in $\ell^1$, does that mean that $(\|x_n\|)$ is Cauchy in $\mathbb F?$
Does $(x_n)$ Cauchy in $\ell^1$ implies $(\|x_n\|_1)$ is Cauchy in $\mathbb F$
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functional-analysis
metric-spaces
banach-spaces