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Homework Help: Sup(X)=-inf(-X) proof.

  1. Oct 28, 2012 #1

    (1) In order to prove that sup(X)=-inf(-X), for XC=R and -X={x E R|-x E X}, does it suffice to write:

    sup(X): for every x E X, x <= M, M E R
    inf(-X): for every x E -X, -x >= m ⇔ x <= -m, m E R

    Hence, m=min(-X) and M=max(X) as they are both part of R and as there could be only one supremum M must be equal to -m.

    Is this a valid proof?

    (2) Which conditions need b to fulfill so that 1/b is bounded, for b E R? I have managed to arrive at |b|>=1. Are there any other requirements, conditions?
  2. jcsd
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