I have been consulting different sources of analysis notes. My confusion comes from these two definitions(adsbygoogle = window.adsbygoogle || []).push({});

\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.

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# Confused by separate definitions of sets which are bounded above

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