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Math Help - Supremum Proof

  1. #1
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    Supremum Proof

    I need help to prove the following question -
    "Let S be a set (S⊂R) and let a be a positive number. Define the set T by T={as: s ∈ S}. Prove supT=asupS" Im not sure how to go about this proof whether it would be using an epsilon style proof or otherwise any help would be appreciated
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  2. #2
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    Hi,
    let S be nonempty and bounded above, then T is also nonempty and bounded above, so both suprema exist. Let's denote q = \mbox{sup }S .
    Is aq an upper bound of T? Yes, because any t \in T can be written as t = as for some s \in S and we have s\leq q and a is positive, so t=as \leq aq.
    Is aq the least upper bound? If it were not, there would be some p \in \mathbb{R} such that p<aq and p is an upper bound of T. Then p/a is an upper bound of S, because for any s\in S we have as \in T so as \leq p and s\leq p/a. Furthermore, p/a < q. But this would mean that q is not the least upper bound of S, which is contradiction.
    We conclude that aq = \mbox{sup }T.
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