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Homework Help: Proving on the completeness theorem of real number

  1. Sep 29, 2011 #1
    1. The problem statement, all variables and given/known data

    (1) ** Show that a function f is continuous at a point c if and only
    if for every sequence (xn) of points in the domain of f such that
    xn ! c we have limn!1 f(xn) = f(c) = f(limn!1 xn).
    (2) Let A be a non-empty subset of R. Dene A := fxjx 2 Ag.
    Show the following statements.
    (a) A has a supremum if and only if A has an inmum, in
    which case we have inf(A) = sup A.
    (b) A has an inmum if and only if A has a supremum, in
    which case we have sup(A) = inf A.
    (3) Show that the completeness axiom of real number system (i.e.
    the Least Upper Bound Property) is equivalent to the Greatest
    Lower Bound Property: Every non-empty set A of real numbers
    that has a lower bound has a greatest lower bound.
    HINT: Use (2).
    (4) * (Monotone Property) Suppose that A B R, where A =6 ;
    and B =6 ;. Show the following statements.
    (a) If B has a supremum, then A has also a supremum, and
    sup A sup B.
    (b) If B has an inmum, then A has also an inmum, and
    inf A inf B.
    (5) ** Let A and B be non-empty subsets of R such that a b for
    all a 2 A and b 2 B. Show that A has a supremum and B has
    an inmum, and sup A inf B

    2. Relevant equations
    The completeness axiom

    3. The attempt at a solution
    I am seriously clueless on how to approach... but I still tried something
    But my method seems a bit weird...

    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
    Last edited: Sep 29, 2011
  2. jcsd
  3. Sep 29, 2011 #2


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