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F g A → Ris defined by (f g)(x) = f(x) g(x) Proposition 212 Suppose that f,g A → R and f ≤ g If g is bounded from above then sup A f ≤ sup A g, and if f is bounded from below, then inf A f ≤ inf A g Proof If f ≤ g and g is bounded from above, then for every x ∈ A f(x) ≤ g(x) ≤ sup A g Thus, f is bounded fromTo get that conclusion, we need to know that f(x) g(y) for all x;y2nd use Proposition 2% " % ( % ) ) / & , 0 % 0 % 1 2 3 4 5 4 6 7 8 9 7 ; 2 ƒXƒgƒŒ[ƒg ƒ~ƒfƒBƒAƒ€ ƒ{ƒu