The inequality is equivalent to $xy + yz + zx \le x + y + z$. Since $x$ is positive and $y \le 1$ we have $xy \le x$. A similar reasoning shows that $yz \le y$ and $zx \le z$. Hence, the inequality holds.

The equality holds if and only if $xy = x$, $yz = y$ and $zx = z$. If $x=0$, then the third equality implies $z=0$ and the second equality then implies $y=0$. Otherwise, we must have $y=1$ and then the second equation implies $z=1$ and from the third equation we get $x=1$. Hence, the equality holds when $x=y=z=0$ or $x=y=z=1$.