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Suppose $X$ is a normed vector space with norm $||\cdot||$. Suppose $S$ is a linear subspace of $X$ and $S$ is closed. Let $x\in X$. I was wondering whether the set $\{||x+s||: s\in S\}$ is closed in $\mathbb{R}$

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  • $\begingroup$ It need not be. $\endgroup$ – Daniel Fischer May 9 '14 at 20:58
  • $\begingroup$ I take a converging sequence $t_n$ in the set. To this sequence there corresponds $s_n$ in $S$, i.e. $t_n=||x+s_n||$. Since $t_n\rightarrow t$ and the norm is continuous then $t=||x+\lim s_n||$. Can I conclude from here that the limit of $s_n$ exists? $\endgroup$ – mathemagician May 9 '14 at 21:02
  • $\begingroup$ No, you cannot. $\endgroup$ – Alex Becker May 9 '14 at 21:02
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    $\begingroup$ See this related question. $\endgroup$ – David Mitra May 9 '14 at 21:04
  • $\begingroup$ @DavidMitra I would go so far as to call this a duplicate of that question. Do you agree? $\endgroup$ – Alex Becker May 9 '14 at 21:08