Show that if $\mathbb{R}^n=W_1\bigcup W_2\bigcup\ldots\bigcup W_k\bigcup\ldots$, where each $W_k$ is a subspace , then $\mathbb{R}^n=W_i$ for some $i$.
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I want to first solve a simple problem as follows. Let $V$ be a finite-dimensional vector space over a field F. If there exist subspaces $W_1$, $W_2$ of $V$ such that $V=W_1\bigcup W_2$, is $V$ either $W_1$ or $W_2$? – Boar Jan 17 '18 at 07:22
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What tools do you have? For example, can you compute the measure of the union? – Elle Najt Jan 17 '18 at 07:24
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@iamokay Which one do you want to solve? The one in the question or the one in the comment? Because the one in the comment is actually false. Its true version (the same with $F$ being an infinite field) cannot be proved as a special case of the one in the question, because the latter is false if you put $\Bbb Q^n$ instead of $\Bbb R^n$. – Jan 17 '18 at 07:27