Finding the upper bound of a product

Question

Let $E=\{\left(x,y,z\right)\} \in \mathbb{R} | x,y,z>0, xy+yz+zx=1 \}$. Prove that there exist $\left(a,b,c\right) \in E$ such that $xyz \leq abc$ for all $\left(x,y,z\right) \in E$.

Attempt. Because $x,y,z$ are positive then they must be bounded. Because $xy+yz+zx=1$.Hence for some $\left(a,b,c\right) \in E$ we have $xyz \leq abc$

Answer 1

$$ xy+yz+zx = 1 \ge \frac{1}{3}(x^2y^2z^2)^{1/3}\Rightarrow xyz \le 3^{3/2} $$

note that $x > 0, y > 0, z > 0$ and $xy+yz+zx = 1$