Solution
= Solution
$E[X_{n+1}\mid\mathcal F_n]=X_n[pc+(1-p)c^{-1}]$. Thus the condition is $pc+(1-p)/c\le1$, or $pc^2-c+1-p\le0$. Since the roots are $(1-p)/p$ and $1$, the range is $(1-p)/p\le c\le1$.
= Solution
$E[X_{n+1}\mid\mathcal F_n]=X_n[pc+(1-p)c^{-1}]$. Thus the condition is $pc+(1-p)/c\le1$, or $pc^2-c+1-p\le0$. Since the roots are $(1-p)/p$ and $1$, the range is $(1-p)/p\le c\le1$.