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This came from my last week's midterm exam problem.

This Course is Advanced Calculus/Introduction to Analysis, and the textbook is Apostol's.
My professor asked us to:

Prove if the claim is believed to be True.

Give counterexample if the claim is believed to be False.

Claim:

Given nonempty subsets $A$ and $B$ of $\mathbb{R}$,
let $C$ = { $x+y:x \in A,\text{ } y \in B$ }.

Then, $\mathit {sup}$ $C$ = $\mathit {sup}$ $A$ $+$ $\mathit {sup}$ $B$

My thoughts:

Since $A$ and $B$ aren't said to be $\pmb{bounded}$, this claim is not, generally, True.

My answer:

False, counterexample given as follow.
let $A$ $=$ $\mathbb{R}$ and $B$ $=$ {$1$}.
Clearly, $\mathit {sup}$ $B$ $=$ $1$.
But, since $\mathbb{R}$ isn't bounded, $\mathit {sup}$ $A$ doesn't exist at all.
Therefore, $\mathit {sup}$ $C$ would not exist, too.
(How could $1$ be added to something that don't exist)

Correct Answer: True

Both TA and Prof. respond that $\mathbb {R}$ itself is bdd above by $+\infty$

My understanding and confusion:

I know and can prove the theorem, which explicitly state that all subsets must have supremum, hence, implicitly implying bounded, and is a bit different from the claim above.

I know for extended $\mathbb{R}$, denoted as $\mathbb{R}^*$ in textbook, $\mathit{sup} \text{ }(\mathbb{R}^*)$ does exist and is $+\infty$

Isn't supremum for any real (sub)set must be a real number?
--- Not saying that supremum must be in this (sub)set, that would be, if exists, maximum.
If True, how can $\infty$ be supremum of $\mathbb{R}$? (Contrast to it being not real, mathematically and per se )

Also, since the answer is True,

does that mean the theorem doesn't actually need "being bounded" as its sufficient condition?

Please enlighten me, appreciate for any help.

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  • $\begingroup$ What is your definition of supremum? $\endgroup$
    – sranthrop
    Commented Nov 17 at 20:03
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    $\begingroup$ One usually allows for $\sup A=\infty$. If this convention is not followed in your class, then you would be right. The course staff's response suggests that this convention is followed in your course. $\endgroup$
    – Andrew
    Commented Nov 18 at 4:00
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    $\begingroup$ That said, @RayW, I would argue neither of the more general tags apply. Nothing in your question deals with the tools introduced to deal with topology or metric spaces in general. The tag system becomes significantly less useful if we apply every tag that sits above the topic. At most general, your question is about ordered spaces, but it's sufficient to only tag it (real-analysis) and (supremum-and-infimum). $\endgroup$
    – Brian Moehring
    Commented Nov 18 at 19:24
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    $\begingroup$ There are some vacuous true/false issues. Is "The Loch Ness monster's diet consisits of every thing the Yeti's diet does" true or false? I'd claim it's either meaningless or vacuously true. What about "The Loch Ness monster eats what the yeti does". I'd say that's more likely false but still... I'd think if a question mentions $\sup A$ as though it exists to presume an assumption is bounded above. And I'd argue that the sentence "doesn't apply" otherwise, rather than is "false". That said though, I don't like the way the question was asked and I don't like your professors answers. $\endgroup$
    – fleablood
    Commented Nov 19 at 16:43
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    $\begingroup$ Actually, your professors are being dishonest. To say $A,B$ are subsets of $\mathbb R$ is tacitly implying your universe is $\mathbb R$ and $\mathbb R$ is certainly not bounded in $\mathbb R$. $\endgroup$
    – fleablood
    Commented Nov 19 at 16:56

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I do not know which definitions have ben given in your class.

Apostol's " Mathematical Analysis" begins by defining the supremum for sets which are bounded above (Definition 1.13). In the sense of this definition you are right: $\sup$ is undefined for sets which are not bounded above.

However, just a little later Apostol introduces the extended real number system $\mathbb R^* = \mathbb R \cup \{ +\infty, -\infty\}$ (Chapter 1.20). He then writes

The principal reason for introducing the symbols $+\infty$ and $-\infty$ is one of convenience. For example, if we define $+\infty$ to be the sup of a set of real numbers which is not bounded above, then every nonempty subset of $\mathbb R$ has a supremum in $\mathbb R^*$. The sup is finite if the set is bounded above and infinite if it is not bounded above.

In this sense your professor is right.

So what is the "correct" answer? It depends of the definition of the supremum.

I can only guess that in your class the sup of a set of real numbers which is not bounded above was defined as $+\infty$, but of course I do not know it. Anyway, Apostol is very clear concerning this point.

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  • $\begingroup$ Thanks for your response, and I am totally aware of this. However, as detailly as I can recall, my prof did give us definition 1.13 while introducing $\mathbb{R}^*$ later but did not define sup $\mathbb{R}$ as $\infty$. $\endgroup$
    – Ray W
    Commented Nov 19 at 9:23
  • $\begingroup$ Also, even if sup $\mathbb{R}$ is defined as $\infty$, and by Apostol's word, it has a sup in $\mathbb{R}^*$, I wonder whether the same set has a sup in $\mathbb{R}$? Since $\infty$ will have a meaning only in the context of $\mathbb{R}^*$, but not in $\mathbb{R}$. For by my understanding of Apostol's context, defining this is simply a process to discuss case in $\mathbb{R}^*$. $\endgroup$
    – Ray W
    Commented Nov 19 at 9:30
  • $\begingroup$ Or there might be something I missed? When $\mathbb{R}$ is defined to have $\infty$ as its sup in $\mathbb{R}^*$, does it simultaneously mean sup $\mathbb{R}$ is $\infty$ in $\mathbb{R}$? $\endgroup$
    – Ray W
    Commented Nov 19 at 9:47
  • $\begingroup$ "When R is defined to have ∞ as its sup in R∗, does it simultaneously mean sup R is ∞ in R?" Absolutely not! Just like $A = \{q\in \mathbb Q|q^2 < 2\}\subseteq \mathbb Q$ but $\sup A \not \in \mathbb Q$. If $\mathbb R$ is bounded in $\mathbb R^*$ (but not in $\mathbb R$) we have $\sup \mathbb R\not \in \mathbb R$. But to be honest, I don't really like your professors. It's not universal to assume our universe is always extended number line. Quite the opposite really. In my opinion you display the potential to eventually become a much better mathematician than they. $\endgroup$
    – fleablood
    Commented Nov 19 at 16:49

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