I need to prove monotonicity assumption on Blackwell’s sufficient conditions for a contraction, that is:
Given the operator T defined as
$(Tf)(x) = sup [F(x,y)+bf(y)]$
I need to show that $f\leq g\Rightarrow Tf\leq Tg$
My first approach is:
$F(x,y) + bf(y)\leq F(x,y) + bg(y)$
$sup F(x,y) + bf(y)\leq sup F(x,y) + bg(y)$
Am I missing something? Is this proof too simple to fully demonstrate the statement above?
Appreciate your help.