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?

Appreciate your help.

  • 2
    $\begingroup$ It seems fine assuming $b\geq 0$ $\endgroup$ Mar 16 '21 at 18:12
  • $\begingroup$ Is your $sup$ over $x$ or $y$? Notations in your first eq. are different from the last eq. $\endgroup$
    – Bertrand
    Mar 18 '21 at 9:11

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