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Math Help - supremum

  1. #1
    Junior Member
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    supremum

    Could someone explain why  \sup\{x_n+y_n\}\leq\sup\{x_n\}+\sup\{y_n\} ?

    Thanks
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  2. #2
    Member Black's Avatar
    Joined
    Nov 2009
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    (Assuming \sup x_n and \sup y_n are finite) Let x=\sup x_n and y=\sup y_n. Then we have

    x_n \le x, \, \forall n \in \mathbb{N}

    y_n \le y, \, \forall n \in \mathbb{N}.

    Add the two together to get

    x_n+y_n \le x+y, \, \forall n \in \mathbb{N}.

    Since x+y is an upper bound for x_n+y_n, it follows that

    \sup(x_n+y_n) \le x+y.
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