$\{x \in \mathbb{R^2} : x_1x_2=1\}$ - Compact

Question

Let A = $\{x \in \mathbb{R^2} : x_1x_2=1\}$.

I try to show that this set in not compact. If I find a sequence for which there is not convergente subsequence, then it will be proved. For the realization, I will use the contrapositive of the Cauchy theorem.

Is anyone could give me a such sequence?

Answer 1

If you want to produce a sequence with no convergent subsequence, let $x_1=1,2,3,4,\dots$.

Answer 2

You note that the graph of the function $y=\dfrac{1}{x}$ is unboundend. For the Heine Borel Theorem a subset of the euclidean plane is compact iff it's closed and bounded.

Answer 3

This set is the graph of the curve $y=1/x$, which is a hyperbola. You should realize that, for example, the sequence $\{ (n,1/n): n\in\mathbb N\}$ lies on this curve but does not contain a convergent subsequence.