Show that the set is not bounded - formal proof - $D = \{(x,y)\in R^2|y=\frac 1 x\}$

Question

is there any way to prove with epsilon \ other way that the set is unbounded?
$$D=\{(x,y)\in R^2\mid xy=1\} = \{(x,y)\in R^2\mid y=\frac 1 x\}$$ Here is a proof my exerciser at class did ( which is not enough proof for me, If there is way to prove with epsilon, I will be glad if there is ):
let $M\in R$, happens: $(M, \frac 1 M )\in D$ Thus:
$||M + \frac 1 M|| = (M^2 + (\frac 1 M)^2)^{(\frac 1 2)} \ge M$
Thus, not bounded. Is there any other proof for it? something with $\epsilon$, any other way which is more friendly?

Answer 1

Notice that : $$\forall n \in \mathbb{N}^*, \left(n, \dfrac{1}{n}\right) \in D$$ and : $$\left\| \left(n, \dfrac{1}{n}\right) \right\|^2 = n^2 + \dfrac{1}{n^2} \to +\infty$$ then $D$ is unbounded.