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After spending a few hours trying to understand Theorem $1.11$ in Rudin's Principles of Mathematical Analysis, I still don't follow the proof.

$1.11$ 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 $\alpha = \inf B$.

This is what I understand so far:

$B$ is bounded below means that $L$ is not empty and $L = \{ y \ | \ y \leq x \ \forall x \in B \}$. Then every $x \in B$ is an upper bound of $L$, which means that $L$ is bounded above. Since $L \subset S$, $L$ not empty, and $L$ is bounded above that implies that $\sup L = \alpha \in S$. And because $\alpha = \sup L$, $\gamma < \alpha$ implies that $\gamma$ is not an upper bound of $L$ and $\gamma \notin B$ since every element of $B$ is an upper bound of $L$.

This is where I get confused:

Since $B$ is bounded below, there exists an $\omega \in S$ such that $\omega \leq x \ \forall x \in B$. Then Rudin claims, "It follows that $\alpha \leq x$ for every $x \in B$." Can someone explain why that is true or at least give me a hint?

Thanks in advance.

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    $\begingroup$ Apologies, but I don't understand why L must be a subset of S. As a counter example, what if B = S, wouldn't some elements of L have to be outside of S? Many thanks! $\endgroup$
    – user68045
    Mar 22, 2013 at 18:17
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    $\begingroup$ @user68045 isn't around and commented over 4 years ago, but: Everything is in $S$ in the context of the theorem. There is nothing indicating some larger ordered set of which $S$ is a subset. If $B=S$ and $B$ is bounded below, then $L$ would consist of the single lower bound which must also be the mimimum element of S. $\endgroup$ Sep 5, 2017 at 3:38

3 Answers 3

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Rewording it slightly: Every element of $B$ is an upper bound for $L$, so if $x\in B$ is less than $\alpha$, then $x$ is an upper bound for $L$ smaller than the least upper bound. This contradicts the definition of least upper bound, so no such $x$ exist. In other words, $\alpha\leq x$ for every $x\in B$.

I do not know why you included a restatement of the fact that $B$ is bounded below. The part you put in quotes follows directly from the last sentence of the previous paragraph in your post:

And because $\alpha = \sup L$, $\gamma < \alpha$ implies that $\gamma$ is not an upper bound of $L$ and $\gamma \notin B$ since every element of $B$ is an upper bound of $L$.

From here you could use contraposition:

$$\gamma<\alpha\implies \gamma\not\in B$$

is equivalent to

$$\gamma\in B \implies \gamma\geq \alpha.$$

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  • $\begingroup$ Your first paragraph cleared up my confusion. Thanks! $\endgroup$
    – Student
    Jan 27, 2012 at 23:04
  • $\begingroup$ I'm a bit confused as to what you are trying to show with the contraposition. Are you trying to show that $\alpha\leq x$ for every $x \in B$? If so how does this achieve that? $\endgroup$ Nov 18, 2017 at 1:21
  • $\begingroup$ @YellowPillow: OP wanted to conclude that $\alpha \leq x$ for all $x\in B$, or equivalently that $x\in B\implies \alpha \leq x$, which is equivalent by contraposition to $\alpha > x\implies x\not\in B$. The latter is what the OP had argued. The way I stated the contraposition aligned with their notation. $\endgroup$ Oct 2, 2018 at 20:04
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I think that Rudin has complicated the proof a little bit. Here is what I learned from his proof to write my own. I think this is more straight forward.

Let $L$ be the set of all lower bounds of $B$ or more precisely $L=\{l\,|\,\forall x:x\in B\implies x\ge l\}$. Since $B$ is bounded below so $L$ is not empty. Also, for every $x$ in $B$ and every $l$ in $L$ we have $l\le x$. This means that every $x\in B$ is an upper bound for $L$. By the L.U.B. property of $S$, we conclude that $L$ has a supremum denoted by $\sup L$. As $\sup L$ is smaller than any other upper bound of $L$ we conclude that $\sup L \le x$ for every $x\in B$. But this means that $\sup L$ is also a lower bound of $B$ and hence $\sup L \in L$. Furthermore, if $l\in L$ is any lower bound of $B$ then $l \le \sup L$. So far, we proved $\sup L$ is a lower bound of $B$ and is greater than any other lower bound of $B$ so it is the greatest lower bound of $B$, that is $\inf B =\sup L$. Accordingly, the set $S$ has the G.L.B. property.

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Rookie's understanding of the proof (maybe helpful for someone in the middle of studying). All corrections much appreciated.

We need to show (1) $\sup L = \alpha$ and $\alpha\in S$ and (2) $\alpha = \inf B$, given the premises.

Just to convince ourselves that some "obvious" parts of the theorem are clear:

If $B\subset S$ is bounded below and $L$ is the set of all lower bounds of $B$, then because $B$ is bounded below (i.e there exists at least one element that is lower or equal to every element of $B$), the set $L$ is not empty.

Now let's derive each enumeration one by one:

(1) Since $L$ is the set of all lower bounds for $B$, any element $x\in B$ is an upper bound of $L$. Therefore, $L$ is bounded above, because all elements of $B$ are upper bounds of $L$. By the premise that set $S$ has L.U.B property and $L$ is a nonempty and bounded subset of $S$, it follows that $L$ has the supremum $\alpha$ (by Definition 1.10) and can be denoted $\alpha := \sup L$. Since $\alpha$ is less or equal to every element of $B$, we have that $\alpha$ is the lower bound for $B$ and thus by the construction of $L$ it is included in $L$, i.e $\alpha\in L$.

Remark: I was confused as to why $L$ has to be a subset of $S$. I understand it as we are given an ordered set $S$ and $B$ is a subset of $S$, bounded below and $L$ is the set of all lower bounds of $B$, it is implicitly implied that $L$ is constructed from the elements of $S$ and thus $L\subset S$. If anyone could elaborate on how they understand it correctly, that would be great.

(2) Take any $\gamma<\alpha$, then $\gamma$ is not an upper bound of $L$. Moreover, it implies $\gamma \notin B$ because all elements of $B$ are an upper bound of $L$ by construction. Now, take any $\beta>\alpha$, then $\beta\notin L$ because $\alpha$ is the upper bound of $L$. We showed that $\alpha$ is the lower bound of $B$ because $\alpha \in L$ and it is the least lower bound because no $\beta>\alpha$ is the lower bound of $B$. Why? Because then $\beta$ would be outside of $L$ which by construction was a set of all lower bounds for $B$.

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