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Let $S,X$ be subsets of $R^n$ given by $$S=\{(a_1,a_2,\dotsc,a_n)\in R^n|\sum a_i^2=1\}$$ $$X=\{(b_1,b_2,\dotsc,b_n)\in R^n|\sum\frac{b_i}{i}=0\}$$

Then prove that $S+X$ is a closed set in $R^n$.

Cameron Buie
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Jack Wislly
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  • The title has very little to do with whatever you are trying to ask (which is not clear, since the notation is horrible). Anyway, what have you tried? – BBBB Aug 22 '13 at 13:50
  • You should check that I haven't changed the meaning of your question in the edit. And I agree with Yuval that you should explain what you already tried and where you got stuck. – mdp Aug 22 '13 at 13:55

2 Answers2

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  1. Prove that the corresponding maps are continuous, hence preimage of closed sets are closed. One of them is actually compact, indeed a circle.

  2. Prove that the sum of a closed set and a compact set is closed, preferably with the limit characterisation.

Marc Palm
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    The sum of closed sets need not be closed. For example, $\bf{Z} + \pi\bf{Z}$ is dense in the real line. It is only true if one of the sets is compact – BBBB Aug 22 '13 at 14:34
  • Thanks, that was too easy. I hope now it's fine. – Marc Palm Aug 22 '13 at 14:37
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Hint: $S$ is $S^{n-1}$ in literature, the $B_1(0)\subset \mathbb{R}^n$ with metric $d(x,y) = \Vert x-y \Vert_2$.
This excercise may well be done using sequences: Let $(a_n)_{n\in\mathbb{N}} \subset S+X$ (pointwise sum?) be a convergent sequence. Then show $$\lim_{n\to\infty} a_n = a \in S+X$$

AlexR
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