This is maybe a very basic question, but I have never seen it done before. Can you use induction to prove that something is not true? In particular if something does not hold in dimension n=1, can I use induction to prove that it does not hold with higher dimension?
2026-03-26 09:38:33.1774517913
Induction to prove that something is not true?
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One example is showing that ${\mathbb R}^n$ is not a union of countably many proper subspaces. For this you can assume the contrary and intersect with a hyperplane and get that ${\mathbb R}^{n-1}$ is a union of countably many proper subspaces. The base case $n = 2$ is easy because there are uncountably many angles through the origin.