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If $A \subset B, a_0 \in A$ and $\beta$ is an upper bound of B then $\sup A \le \beta.$

$\textbf{Proof:}$

Since $a_0 \in A,$ then $A \ne \emptyset,$ thus $\sup A$ exists. Since $A \subset B,$ then $\sup B$ is an upper bound of $A$. By definition, $\sup A$ is the least upper bound of $A.$ This tells us that $\sup A \le \sup B.$ Also, since $\beta$ is an upper bound of $B,$ then $\sup B \le \beta.$ This implies $\sup A \le \sup B \le \beta,$ hence $\sup A \le \beta.$

Please tell me if this is a correct way in proving such a statement. Any constructive criticism would be much appreciated. Thank you very much.

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    $\begingroup$ yes you are correct $\endgroup$
    – Learnmore
    Feb 21, 2015 at 6:44
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    $\begingroup$ Awesome! thank you!!! $\endgroup$
    – mathamphetamines
    Feb 21, 2015 at 6:45
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    $\begingroup$ Do you allow $+\infty$ as a value for $\sup A$, or does the definition require this to be a real number? If the latter, then the first sentence in your proof is unjustified. If the former, you should probably clarify to avoid misunderstandings. $\endgroup$
    – Andrés E. Caicedo
    Feb 21, 2015 at 7:03
  • $\begingroup$ @ Andres Caicedo I believe in Rudin, the $sup \; A$ is allowed to be $+\infty$ $\endgroup$
    – mathamphetamines
    Feb 21, 2015 at 7:06
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    $\begingroup$ You can use $\sup A$, which renders as $\sup A$. It looks a bit better and, most importantly, it has correct spacing. (It is not all lumped together as in $sup A$ $sup A$.) $\endgroup$
    – Martin Sleziak
    Feb 21, 2015 at 8:32

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I agree. It seems fine. But I'm wondering if you can't just say that, if $\beta$ is an upper bound for $B$ and $A\subseteq B$, then $\beta$ is an upper bound for $A$ and hence $\sup A\leq\beta$. Maybe we have to be extra careful because of the versions of the definitions we have in mind? Saying that $A\subseteq B\Rightarrow \sup A \leq \sup B$ seems like a stronger thing to use in your reasoning than just $A\subseteq B\Rightarrow$ "an upper bound for $B$ is an upper bound for $A$" and "$\beta$ is an upper bound for $A$" $\Rightarrow \sup A \leq \beta$. Think about it and let me know what you think. :)

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  • $\begingroup$ Ah. This way of thinking about it might also resolve Andres' concern. $\endgroup$
    – Michael Cotton
    Feb 21, 2015 at 7:07
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    $\begingroup$ I'm a beginner when it comes to proofs and especially real analysis so I am unsure. I do appreciate the time you're taking to discuss this though. I mean mathematics needs to be exact after all. $\endgroup$
    – mathamphetamines
    Feb 21, 2015 at 7:09

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