Confused by separate definitions of sets which are bounded above

[jacksonjs20]

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.

 

[mathman]

The first definition says an M > 0 exists. It doesn't preclude an M < 0 being a bound.

 

[jacksonjs20]

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?

 

[mathman]

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