Solution:

Note that $x$, $y$, and $z$ are the roots of the polynomial
$$(t - x)(t - y)(t - z) = t^{3} - (x + y + z) t^{2} + (x y + x z + y z) t - x y z.$$
We claim that this polynomial has no negative roots. To see this, if we plug in a negative value of $t$, all four terms we are adding are negative, so the sum is negative. If we plug in $t = 0$, the first three terms are negative, and $-x y z$ is negative since $x y z$ is positive, so the sum is again negative. Thus, whenever $t \leq 0$, the sum is negative, so any real root must satisfy $t > 0$. So, since $x$, $y$, and $z$ are all roots, they must all be positive.