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I have the set

$$A = \{-n + 1/n \ : n \in \mathbb{N}\}$$

My attempt

I tried to find some limit point in A, but

$$ \lim_n (-n + 1/n) = -\infty $$

Is there anyone to help?

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    $\begingroup$ Can you show that $A$ has only finitely many elements in any bounded set? $\endgroup$ – Lord Shark the Unknown Oct 13 '18 at 7:03
  • $\begingroup$ @LordSharktheUnknown bounded subset? or $A \cap B$ is finite, and B is a bounded set? $\endgroup$ – user198044 Oct 13 '18 at 7:04
  • $\begingroup$ I have this suspicion that $A' = \emptyset \subseteq A$ ? If I'm right, then it can be proved if $\forall n \in \mathbb N, -n + \frac1n$ is isolated. $\endgroup$ – user198044 Oct 13 '18 at 7:04
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The complement of $A$ is $\bigcup_{n\in\mathbb N}(-n-1+\frac1{n+1},-n+\frac1n)\bigcup(0,+\infty)$, which is a union of open intervals, thus open. Therefore $A$, as the complement of an open subset, is closed.


Hope this helps.

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If you want to prove that your set is closed using sequences, you can use the fact that, precisely because $\lim_{n\to\infty}-n+\frac1n=-\infty$, the only convergent sequences of numbers of the firm $-n+\frac1n$ are those that are constant after a certain point. And every such sequence converges to another element of your set, obviously.

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The intersection of the set with any compact subset (i.e. closed and bounded) of $\mathbb{R}$ is finite hence closed. As metrics spaces are $k$-spaces, the set is closed.

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