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Math Help - Sequences- Supremum

  1. #1
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    Sequences- Supremum

    Is it true that for every two sequences
    {xn} n goes from 1 to infinityand {yn}n goes from 1 to infinity satisfying for every n xn < yn one has

    a. sup xn <or=sup yn?
    b. sup xn < sup yn?

    How do you prove or disprove these?
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  2. #2
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    Quote Originally Posted by amm345 View Post
    Is it true that for every two sequences









    {xn} n goes from 1 to infinityand {yn}n goes from 1 to infinity satisfying for every n xn < yn one has

    a. sup xn <or=sup yn?
    b. sup xn < sup yn?

    How do you prove or disprove these?

    \forall n \in \mathbb{N}\,,\,\, \frac{n-1}{n}<1\,\,,but\,\, \sup \left\{\frac{n-1}{n} \right\}_{n=1}^\infty=\sup \left\{1,1,...\right\}

    Tonio
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    Can you please elaborate on that?
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  4. #4
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    He gave an example of sequences \{x_n\} and \{y_n\} with x_n< y_n but [tex]sup \{x_n\}= sup \{y_n\}[/itex]

    Perhaps a simpler example would be x_n= -\frac{1}{n} and y_n= 0 for all n. x_n< y_n for all n but their supremums (in this case they are both non-decreasing sequences so their supremums are just their limits- and both are 0.
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