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Denote V {0} by V Observe that a non-empty linearly independent subset of V is an element of
P(V ) . Let L P(V ) {} be the set of non-empty linearly independent subsets of V and partially
order L by inclusion
(a) Let C be a chain in L. Prove that U= UUC U is an upper bound for C.
(b) Let M be a maximal element of L prove that M spans V .
(c) Now appeal to Zorn’s lemma to prove that V has a basis.
Solution
Suppose a partially ordered set P has the property that every chain has an upper bound in P. Then
the set P contains at least one maximal element.

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