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April 16, 20260 citationsOpen Access

Completeness of Rm

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DSDr. BS Satpute

Key Points

  • The research investigates the properties of bounded sequences, specifically regarding the existence of monotone subsequences and their convergence.
  • Utilized the result that every sequence of real numbers contains a monotone subsequence.
  • Analyzed bounded sequences with unequal lim sup and lim inf.
  • Provided proofs of the bounded sequence convergence in Euclidean spaces.
  • Established that bounded sequences with unequal lim sup and lim inf possess at least two disjoint monotone subsequences.
  • Confirmed that every bounded sequence in R has a convergent subsequence, aligning with the BW theorem.
  • Demonstrated the completeness of Euclidean spaces under the usual metric.

Abstract

Using the result ‘every sequence of real numbers has a monotone subsequence’ we prove: If a bounded sequence has unequal lim sup and lim inf, then it has at least two monotone (convergent) subsequences whose range sets are disjoint; (BW theorem) Every bounded sequence has a convergent subsequence, and R is a complete space under usual metric on R. From these we obtain easy proofs of (BW theorem): Every bounded sequence in a Euclidean space has a convergent subsequence, and every Euclidean space is complete.

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Cite This Study

Dr. BS Satpute (2026) studied this question.

synapsesocial.com/papers/69e07dfe2f7e8953b7cbeefbhttps://doi.org/10.5281/zenodo.19117657
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