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Thread: supremum

  1. #1
    Junior Member
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    supremum

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

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

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

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

    Add the two together to get

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

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

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