(adsbygoogle = window.adsbygoogle || []).push({}); Maximal subspace

Problem: Prove that every vector space V has maximal subspace, i.e. a proper subspace that is not properly contained in a proper subspace of V.

I let A be the collection of all proper subspaces of V, but I can't prove that every totally ordered subcollection of A has an upper bound in A. The problem that the union of proper subspaces is not necessarily a proper subspace of V. What do I do now?

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# Basis of a vector space

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