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Math Help - Upper Bound

  1. #1
    MHF Contributor harish21's Avatar
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    Upper Bound

    S = {x | x in R , x>=0, x^2 < c}

    a. Show that c+1 is an upper bound for S and therefore, by completeness axiom, S has a least upper bound that we denote by b.

    b. Show that if b^2>c, we can show choose a suitably positive number r such that b-r is also an upper bound for S, thus contradicting the choice of b as an upper bound.

    c. If b^2< r, then we can choose a suitable positive number r such that b+r belongs to S, thus contradicting the choice of b as an upper bound of S.

    All I could do here was to show that
    if c<=1, then x<1<1+c. So 1+c is an upper bound for S
    and
    if c>1, x<Sqrt(c)<c<1+c. So 1+c is an upper bound for S.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by harish21 View Post
    S = {x | x in R , x>=0, x^2 < c}

    a. Show that c+1 is an upper bound for S and therefore, by completeness axiom, S has a least upper bound that we denote by b.

    b. Show that if b^2>c, we can show choose a suitably positive number r such that b-r is also an upper bound for S, thus contradicting the choice of b as an upper bound.

    c. If b^2< r, then we can choose a suitable positive number r such that b+r belongs to S, thus contradicting the choice of b as an upper bound of S.

    All I could do here was to show that
    if c<=1, then x<1<1+c. So 1+c is an upper bound for S
    and
    if c>1, x<Sqrt(c)<c<1+c. So 1+c is an upper bound for S.
    (1+c)^2=1+2c+c^2\geqslant c^2
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