Consider the two-component mixture $$ F(z)=\lambda F_1(z)+(1-\lambda)F_2(z) $$ where all the $F$'s are CDFs and $\lambda\in [0,1]$.
A1: Assume that $F(z)=0$ $\forall z\leq 0$.
A1 implies that $F(\cdot)$ is not compatible with
$F_1(\cdot)$ such that $F_1(z)>0$ for some $z\leq 0$ and $\lambda>0$.
(In light of the previous bullet point) $F_1(\cdot)$ and $F_2(\cdot)$ both Logistic. This is because the Logistic CDF is zero only in the limit for any value of the scale and location.
Question: is the claim correct?