I have been trying to solve the following problem: Let $M \subset \mathbb R^3$ be the set of points $(x,y,z) \in \mathbb R^3$ at which $xy + xz + yz = 1.$ Prove that $M$ is a $2$-dimensional manifold. Prove that $M$ is not compact.
I have so far proven that $M$ is a $2$-dimensional manifold by checking for singularities, but am unsure how to prove that $M$ is compact. Is it sufficient to prove that $M$ is unbounded because $(x,y,z)$ is unbounded (at least as far as I can see) and hence not compact? Any help would be greatly appreciated.
For $n\in\mathbb{N}$, let $x_n = n$ and $y_n = \frac{1}{2n}$. Then $$xy + xz + yz = \frac{1}{2} + z\frac{2n^2+1}{2n},$$ so that by taking $z_n=\frac{n}{2n^2+1}$, the points $(x_n,y_n,z_n)$ are on the manifold. This shows that the manifold is not bounded (as a subspace of $\mathbb{R}^3$), and thus not compact.