# Example of closed, non bounded set in R^2

I am supposed to give an example of a closed set that is not bounded in $\mathbb{R}^2$. My idea was the graph of $y=1/x, \forall x$. If I take the complement of it, I get an open set. So the graph of $1/x$ is closed, but not bounded. But I am not sure of it. Could you please elaborate on it and give me a clue how to approach?

Thanks in advance!

• There's $\mathbb{R}^2$ itself; or, say, the $x$-axis. Your idea works as well. – Jordan Oct 3 '14 at 5:37

## 2 Answers

Hint: Let $(x_n,y_n)$ be convergent sequence of elements such that $(x_n,y_n) \in Graph$. Prove that $\lim (x_n,y_n) \in Graph$

• Should i take a specific graph? @jonasgomes – Marion Crane Oct 3 '14 at 5:37
• Graph in this case is the set $(x,\frac{1}{x})$, the graph of the function you suggested. – Jonas Gomes Oct 3 '14 at 5:37

An easier way to think (at least for me) is to consider a bunch of isolated points. They must form a closed set as a set of isolated points contains all (there are none) its limit points. So you could just pick an unbounded set of isolated points.