A short proof of completion theorem for metric spaces

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The completion theorem for metric spaces is always proven using the space of Cauchy sequences. In this paper, we give a short and alternative proof of this theorem via Zorn’s lemma. First, we give a way of adding one point to an incomplete space to get a chosen non-convergent Cauchy sequence convergent. Later, we show that every metric space has a completion by constructing a partial ordered set of metric spaces.

Completion theorem, metric space, complete space, zorn's lemma

Короткий адрес: https://readera.org/147232863

IDR: 147232863   |   DOI: 10.14529/mmph210209

Список литературы A short proof of completion theorem for metric spaces

  • Lusternik, L.A. Elements of functional analysis / L.A. Lusternik, V.I. Sobolev. - Hindustan Publishing Corp., Delhi and Halsted Press, New York, 1974. - 360 p.
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