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I'm having trouble figuring out where to go with this problem. Any hints or strategies would be appreciated. I have just the basic definitions for open sets, distance metrics, etc.

Consider $\Bbb R^\infty=\left\{(a_n):\sum_{n=1}^\infty a_n^2<\infty\right\}$ with the metric $$d\big((a_n),(b_n)\big)=\left[\sum_{n=1}^\infty(a_n-b_n)^2\right]^{1/2}\;.$$ Let $A=\left\{(a_n):|a_n|<\frac1n\text{ for all }n\right\}$, and $A\subseteq\Bbb R^\infty$.

  1. Show that for every $\epsilon>0$, $B(\underline 0,\epsilon)\cap A^c\ne\varnothing$ where $\underline 0$ represents the sequence $(0,0,0,0,\ldots)$.
  2. Deduce that $A$ is not an open set in $\Bbb R^\infty$.
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  • $\begingroup$ So, what is your question? $\endgroup$
    – Pedro
    Oct 17, 2014 at 1:06

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HINT: Fix $n\in\Bbb Z^+$. Consider the sequence whose terms are all $0$ except for the $n$-th term, which is $\frac1n$. Is this sequence in $A$? How far away from $\underline 0$ is it?

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