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Math Help - infimums and supremums of Functions

  1. #1
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    infimums and supremums of Functions

    Let f and g be bounded functions from a nonempty set X into R.

    Prove that if f(x) <= g(x) for all x in X, then inf f(X) <= inf g(X) and
    sup f(X) <= sup g(X).
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  2. #2
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    Quote Originally Posted by Jdg6057 View Post
    Let f and g be bounded functions from a nonempty set X into R.

    Prove that if f(x) <= g(x) for all x in X, then inf f(X) <= inf g(X) and
    sup f(X) <= sup g(X).
    Assume supf(X)> supg(X)........................................... ................................................1

    From the definition of the supremum we have:

    for all x , if xεΧ then  f(x)\leq supf(X).................................................. ...........................................2

    for all x, if xεΧ then  g(x)\leq supg(X).................................................. ............................................3

    AND

    for all ε>0 and hence for ε = supf(X)-supg(X)>0 there exists a yεf(X) such that :

    supf(X)-(supf(X)-supg(X)< y \leq supf(X) or


    supg(X) <y \leq supf(X).................................................. ...........................................4



    But since y belongs to f(X) y= f(x) and xεΧ and since for all xεΧ , f(x)\leq g(x) (4) becomes:

    supg(X)< g(x).............................................. ........................................5


    And using (3) we end up with the contradiction:

    supg(X) < supg(X)

    the proof for the infemums is a similar one
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