absolutely summable series have Cauchy partial sums

[[concept]]

Theorem

Let $V$ be a vector space. If $\{{\sum }_n{v}_n{\}}_n\subset V$ is absolutely summable, then the sequence of partial sums ${\left\{{\sum }_{n=1}^m{v}_n\right\}}_{m=1}^{\infty }$ is Cauchy.

Proof Exercise. Hint: first prove for$V=\mathbb{R}$

Mentions

Mentions

TABLE file.mday as "Last Modified"
FROM [[]]
 
FLATTEN choice(contains(artist, this.file.link), 1, "") + choice(contains(author, this.file.link), 1, "") + choice(contains(director, this.file.link), 1, "") + choice(contains(source, this.file.link), 1, "") as direct_source
 
WHERE !direct_source
SORT file.mday ASC
SORT file.name ASC

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	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}