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In Rudin $1.11$ Theorem Proof he claims the following

Theorem. Suppose $S$ is an ordered set with the least upper bound property $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of all lower bounds of $B$. Then $$\alpha = \sup L$$ exists in $S$, and $\alpha = \inf B$.

Proof. Since $B$ is bounded below, $L$ is not empty. Since $L$ consists of exactly those $y \in S$ which satisfy the inequality $y \leq x$ for every $x \in B$, we see that every $x \in B$ is an upper bound of $L$. Thus $L$ is bounded above. Our hypothesis about $S$ implies therefore that $L$ has a supremum in $S$ call it $\alpha$

If $\gamma < \alpha$ then $\gamma$ is not an upper bound of $L$, hence $\gamma \notin B$. It follows that $\alpha \leq x $ for every $x \in B$. Thus $\alpha \in L$

If $\alpha < \beta$ then $\beta \notin L$ since $\alpha$ is an upper bound of $L$

We have shown that $\alpha \in L$ but $\beta \notin L$ if $\beta > \alpha$. In other words, $\alpha$ is a lower bound of $B$, but $\beta $ is not if $\beta > \alpha$. This means that $\alpha = \inf B$

I am confused in the following:

I don't follow why $L \subset S$ given $S$ is an ordered set with the least upper bound property and $B \subset S$, $B$ is not empty and $B$ is bounded below.

If $L$ is not a subset of $S$, then the assumption of the proof will not follow, I think I have missed something.

Can someone help me out? This proof is in Rudin's analysis page 5

Thanks

Edit for clarification

Suppose the following let $S = (0, x]$, for which $x $ is some real positive number, we know $S$ is an ordered set with the least upper bound property, let $B = (0, y]$ for which $y < x$ and $y$ is positive real number, then $L = (-\infty, 0]$, we note that $\inf B = \sup L = 0$ however $0 \notin S$, thus we proved that an order set $S$ with the least upper bound property with $B = (0, y] \subset S \Rightarrow \inf B \notin S$

Hosein Rahnama
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Kevin
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  • $S$ is the "universe" here. $L$ is by definition a subset of $S$ – David P Dec 21 '14 at 21:04
  • What else do you think would be in $L$ besides members of $S$? – Tim Raczkowski Dec 21 '14 at 21:05
  • Sorry if this question is elementary, but by the definition of ordered set, an ordered set is only a set in which order is defined, and order is defined by definition 1.5, I think you can have a set that is ordered without it being a universal set. I didn't know that all ordered set is the universal set. – Kevin Dec 21 '14 at 21:07
  • Had the question said $\mathbb{R}$ instead of $S$, would you argue that maybe $L$ is a subset of $\mathbb C\setminus \mathbb{R}$? – David P Dec 21 '14 at 21:13
  • If the theorem noted that $S$ is $\mathbb{R}$ or $\mathbb{C}$ I would not argue that $L$ is not in real or complex, but the theory seems to imply that $S$ is an arbitrary ordered set with least upper bound property, so I am a bit confused – Kevin Dec 21 '14 at 21:16
  • $S$ is an arbitrary linearly ordered set with the least upper bound property. It’s also clearly the set in which you’re working, so $L$ is implicitly a subset of it. There is no real difference between working with $S$ and working with $\Bbb R$, say. – Brian M. Scott Dec 21 '14 at 21:27
  • You are imagining you have a lonesome set $S$ with a particular structure, which aside from its own subsets, exists independently of any other set in existence. So any lower bounds of $B$ must belong to $S$. For example, given the set of natural numbers with the standard ordering and $B={2,3,4,...}$ the set of all lower bounds of $B$ is just ${1,2}$. $-17$ is not a lower bound of $B$ here, since we don;t care about things which aren't in $\mathbb{N}$. Maybe the integers haven't even been defined. – David P Dec 21 '14 at 21:29
  • Sorry I still don't quit get it, I am thinking if I have a set $S =\mathbb{R}$ and you take out the negative part, and $0$, you would have a set with no $\inf S$ since $0 = \inf S$, maybe my construct of set is not clear. – Kevin Dec 21 '14 at 21:35
  • If $S=\mathbb R$ and $B=(0,\infty)$ then $L=(-\infty,0]\subset S$. $\text{inf}B = \text{sup}L=0$ – David P Dec 21 '14 at 21:38
  • if $S = (0, x]$ for which $x$ is some real positive number, then $S \subset \mathbb{R}$ is non empty and have an upper bound, thus it must have least upper bound property. – Kevin Dec 21 '14 at 21:48
  • just to clarify, let $S = (0, x]$, for which $x $ is some real positive number, we know $S$ is an ordered set with the least upper bound property, let $B = (0, y]$ for which $y < x$ and $y$ is positive real number, then $L = (-\infty, 0]$, we note that $\inf B = \sup L = 0$ however $0 \notin S$ – Kevin Dec 21 '14 at 22:03
  • @Kevin: No, your $S=(0,x]$ does not have the lub property: every element of $S$ is an upper bound for the subset $\varnothing$ of $S$, which therefore does not have a least upper bound. – Brian M. Scott Dec 21 '14 at 22:10
  • Oh that is very interesting, I didn't know $\emptyset$ is ordered. – Kevin Dec 21 '14 at 22:20
  • @Kevin: It doesn’t matter whether $\varnothing$ is ordered: it’s a subset of $S$, so if every subset of $S$ has a least upper bound in $S$, then $\varnothing$ must have a least upper bound in $S$. – Brian M. Scott Dec 22 '14 at 01:04
  • So what is the least upper bound of $\emptyset$ ? – Kevin Dec 22 '14 at 05:38
  • @Kevin $\sup(\emptyset) = \min(S)$ by definition. All elements of $S$ are upperbounds. – Henno Brandsma Dec 22 '14 at 09:26
  • BTW, your $B$ example (from the last part) is not bounded below (in $S$, which is the point!). So we need not show that $B$ has an infimum to see that $S$ has the largest lower bound property . – Henno Brandsma Dec 22 '14 at 09:28

4 Answers4

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The proof is fine. You just need to realise that everything "lives" in $S$.

So $(S,<)$ is linearly ordered and satisfies the lub property. This means that every $B \subseteq S$ that is bounded above (which means: $\exists b \in S: \forall x \in B: x \le b$) then $B$ has a least upper bound. Now he wants to prove that $S$ has the glb-property. So for every $B \subseteq S$, if $B$ has a lower bound (so $\exists b \in S: \forall x \in B: b \le x$) there exists a greatest lower bound for $B$.

So if we have such a $B$ that is non-empty and bounded below by definition of being bounded below the set $L = \{b \in S: \forall x \in B: b \le x \}$ is non-empty. This is what being bounded below means in the ordered set $S$. And as $B$ is non-empty, pick $x \in B$. Then for every $b \in L$, by definition of being in $L$: $b \le x$. So $x$ (which is in $B \subseteq S$) shows that $L$ is bounded above (in $S$), and the rest of the proof goes through.

In your example, $S = (0, 2]$ and $B = (0,1]$ (for definiteness) in their usual order, the $S$ satisfies the lub-property, but the $B$ is not bounded below in $S$ (For every $x \in S$ , with $x < 2$, $\frac{x}{2} < x$ and lies in $B$. So $x$ is not a lower bound for $B$.). So we don't have to show that $B$ has a greatest lower bound, as it has no lower bound at all. So the example is irrelevant. It's not a counterexample to $S$ also having the greatest lower bound property.

Scientifica
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Henno Brandsma
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  • I do not understand how \textbf{S} being universal is implied? It seems odd that this is not clarified...I spend a good hour struggling – Zero Apr 11 '17 at 14:01
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    @Zero everything lives in $S$ because the theorem starts by suppose $S$ is a linearly ordered set etc. – Henno Brandsma Apr 11 '17 at 14:44
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Proof. Suppose that a nonempty set $A$ has a lower bound, call it $ℓ$. Define $L$ as the set of all lower bounds of $A$, then $L$ is nonempty ($ℓ$ ∈ $L$). Observe that each member of the nonempty set A is an upper bound of $L$ so by the least upper bound property, $L$ has a least upper bound. Call this element $α$. First observe that $α$ is a lower bound for $A$. Otherwise, there exists an element $b$ ∈ $A$ with $b$ < $α$, but each element of $A$ is an upper bound for $L$, so this element $b$ is an upper bound of $L$ which is smaller than $α$, the least upper bound of $L$. This would be a contradiction. Therefore, $α$ ∈ $L$. Also, if $ℓ$ is any lower bound of $A$, then $ℓ $≤ $α$since $$α = sup L$$. Hence α is the greatest lower bound of $A$.

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If you look back on Definition 1.7, the lower/upper bound is defined as belonging to $S$. Thus, $L\subset S$ by definition.

Leland Stirner
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  • This is absolutely right. Notice that Definition 1.7 states that when a set E is bounded above, then the upper bound is in S. Eventually this works in the same way for lower bounds. Thus, when saying that B is bounded below and that L is the set of all lower bounds of B, it is necessarily the case that all elements of L are in S. – Alex Dec 24 '21 at 00:55
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Same question here, but I think one of the above response is right in saying that the bounds of a bounded subset E is in S.

See Definition 1.7, it defines the "boundedness" of a subset in S. It says that for a subset E to to "bounded", the upper/lower bound must be in S. Therefore the lower/upper bounds of subset E must be in S.

Juliet
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