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A vector space over $R$ is not a countable union of proper subspaces

How to prove that any vector space(of any dimensions) over an infinite field can not be written as finite union of its proper Subspaces? I do not want to use induction.

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    Your previous questions have been answered, but you haven't accepted any of the answers. Were none of them acceptable? Please note that accepting answers serves not only to acknowledge the effort others have put into answering your questions, but also to mark the questions as answered so that they don't keep floating around in the system as unanswered questions and people keep coming back to them.2011-11-18

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