Show $\{-n + 1/n \ : n \in \mathbb{N}\}$ is a closed set

Question

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?

Answer 1

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.

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Hope this helps.

Answer 2

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.

Answer 3

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.