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?

  • 1
    $\begingroup$ 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

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.


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.


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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