Wikipedia:Reference desk/Archives/Mathematics/2021 May 11

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May 11

Bolzano–Weierstrass theorem

Does this theorem "Each bounded sequence in R^n have a convergent subsequence" assume that the sequence is an infinite sequence. Because I looked at the Wikipedia article for sequence and it could be either finite or infinite, but the theorem doesn't seem to be true if it is just finite sequence. So why is it not specified "each infinite sequence..."? Nikolaih^☎️📖 22:29, 11 May 2021 (UTC)

Yes, that's meant to be implicit. Like many mathematical terms, "sequence" can mean different things in different contexts, but in analysis it generally means an infinite sequence.--2404:2000:2000:8:B434:8133:83DF:8A34 (talk) 22:41, 11 May 2021 (UTC)

Specifically, the term "bounded sequence" is used only for infinite sequences.  --Lambiam 06:11, 12 May 2021 (UTC)

...since all finite sequences in R are bounded anyway, tho of course not in extended reals. 95.168.118.77 (talk) 17:29, 14 May 2021 (UTC)

The most reasonable way to extend the concept to other domains is, I think, the following. Let $S$ be some preordered set. Then an infinite sequence $a:\mathbb {N} \to S$ is bounded if there exist $L,M\in S$ such that $L\leq a_{n}\leq M$ for all $n\in \mathbb {N}$. For the domain $S=\mathbb {R} \cup \{-\infty ,+\infty \},$ this means that all sequences are bounded, witnessed by taking $L=-\infty ,M=+\infty .$  --Lambiam 08:47, 16 May 2021 (UTC)