# Proof monotonicity on Blackwell sufficient conditions

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:

$$F(x,y) + bf(y)\leq F(x,y) + bg(y)$$

$$sup F(x,y) + bf(y)\leq sup F(x,y) + bg(y)$$

$$T(f)(y)\leq T(g)(y)$$

Am I missing something? Is this proof too simple to fully demonstrate the statement above?

• It seems fine assuming $b\geq 0$ Mar 16 '21 at 18:12
• Is your $sup$ over $x$ or $y$? Notations in your first eq. are different from the last eq. Mar 18 '21 at 9:11