HW01 - Modern Analysis 1 Homework 01 1. Let F be an ordered...

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Unformatted text preview: Modern Analysis 1 Homework 01 1. Let F be an ordered field. Let A and B be subsets of F for which sup A and sup B exist (in F ); define subsets A + B and AB of F by A + B = {a + b : a ∈ A, b ∈ B }, AB = {ab : a ∈ A, b ∈ B }. (i) Does sup(A + B ) exist in F ? If so identify it; if not, show why. (ii) Does sup(AB ) exist in F ? If so identify it; if not, show why. Solution For convenience, write α = sup A and β = sup B . (i) If a ∈ B and b ∈ B then a α and b β (because α and β are upper bounds) whence (as noted in class) a + b α + β ; this shows that α + β is an upper bound for A + B . Now let u ∈ F be any upper bound for A + B . If a ∈ A and b ∈ B are arbitrary, then u a + b so that u − b a; as a here is arbitrary, we deduce that u − b is an upper bound for A so that u − b α (because α is least). Rearrange to obtain u − α b and deduce similarly that u − α β . Finally, the conclusion u α + β shows that α + β is the least upper bound for A + B . (ii) In general, the set AB need not be bounded above: one of A or B may contain a negative element while the other contains ‘arbitrarily large’ negative elements. If A and B are also bounded below, then AB will be bounded above (but its supremum, if it exists, need not be αβ ; why?). However, if A and B contain only positive (or rather, nonnegative) elements, then sup(AB ) exists and equals αβ ; simply replace differences by quotients in the argument for part (i). 1 ...
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