Solution
= Solution
Since $G_X(1)=1$ and $G_X'(1)=\mu>1$, for $t<1$ sufficiently close to one,
$$
G_X(t)-t<0.
$$
At zero, $G_X(0)-0\geq0$. Continuity therefore gives a fixed point in $[0,1)$, and part (b) implies $d<1$.
= Solution
Since $G_X(1)=1$ and $G_X'(1)=\mu>1$, for $t<1$ sufficiently close to one,
$$
G_X(t)-t<0.
$$
At zero, $G_X(0)-0\geq0$. Continuity therefore gives a fixed point in $[0,1)$, and part (b) implies $d<1$.