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Confused by separate definitions of sets which are bounded above

  1. Aug 23, 2011 #1
    I have been consulting different sources of analysis notes. My confusion comes from these two definitions

    \begin{defn} Let S be a non-empty subset of $\mathbb{R}$.
    \begin{enumerate}
    \item $S$ is Bounded above $ \Longleftrightarrow\exists\,M > 0$ s.t. $\forall\, x\in S$, $x\leq M$.
    \item $S$ is Bounded above $ \Longleftrightarrow\exists\,M\in\mathbb{R}$ s.t. $\forall\, x\in S$, $x\leq M$.
    \end{enumerate}
    \end{defn}

    My question: Why in the first definition does M have to be strictly greater than 0?

    e.g.If we consider the set S :={-3,-2,-1}

    Then is S bounded above by -1?. I know that S is bounded above by all positive integers. Though, M = -1 appears to be a suitable choice of M in the second definition, to satisfy S being bounded above, but not the first.

    I have thoroughly confused myself over this matter and would be grateful for any insight into the matter. Thanks in advance.
     
  2. jcsd
  3. Aug 23, 2011 #2

    mathman

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    The first definition says an M > 0 exists. It doesn't preclude an M < 0 being a bound.
     
  4. Aug 23, 2011 #3
    Thank you for your prompt reply. That is exactly the response i was looking for. Just to clarify
    The existence of an m > 0 is a necessary condition for a set to be bounded above. Though, provided it exists, an m < 0 which is an upper bound is sufficient to justify that a set is bounded above.

    Is this correct?
     
  5. Aug 24, 2011 #4

    mathman

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    No: existence of any m is necessary and sufficient - in fact, it is definition of bounded. The sign is irrelevant.
     
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