Connected Subset of Real Numbers: Bounded Below, Unbounded Above

  • Thread starter tarheelborn
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In summary, if S is a connected subset of R that is bounded below but not above, then S is either equal to [a, ∞) or (a, ∞) for some a∈R. This can be proven by showing that if a and x are two points in S, then all points in between them are also in S, and that a is either in S or approaching it. This conclusion is based on the fact that the only connected subsets of R are intervals.
  • #1
tarheelborn
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If [tex]S[/tex] is a connected subset of [tex]\mathbb{R} [/tex] and [tex]S[/tex] is bounded below, but not above, then either [tex]S=[a, \infty)[/tex] or [tex]S=(a, \infty)[/tex] for some [tex]a \in \math{R}[/tex].
 
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  • #2
What have you tried so far? Do you know what connected subsets of R look like?
 
  • #3
Yes I am familiar with the types of connected subsets of [tex] \mathbb{R}[/tex]. I know that if [tex]a<b \in S[/tex] then [tex] [a,b] \subseteq S[/tex]. I think I need to find some point in [tex] [a,b][/tex] and show that [tex]a[/tex] is the least greatest lower bound of that interval whether or not [tex]a[/tex] is included in the interval. But I am not sure how to start.
 
  • #4
I see that if a is the glb for S, then a is either in S or s is approaching a for s in S. And I see that there is no upper bound, so the right end of the interval is infinity. But I need help on writing that up formally. Please. Thank you!
 
  • #5
tarheelborn said:
Yes I am familiar with the types of connected subsets of [tex] \mathbb{R}[/tex]. I know that if [tex]a<b \in S[/tex] then [tex] [a,b] \subseteq S[/tex].

More specifically than just this, the only connected subsets of R are intervals
 
  • #6
I know. I must not be stating my problem clearly. I need to prove that a connected subset of R that is bounded below but not above is equal to either [tex][a, \infty)[/tex] or [tex](a, \infty)[/tex] specifically. I am supposed to use a lemma that says if two points are in a connected subset of the real numbers, then all points in between these two points are also in the subset.
 
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  • #7
I have gotten this far with the proof:


Since [tex]S[/tex] is bounded below, [tex]S[/tex] has a greatest lower bound, say [tex]a[/tex]. Since [tex]S[/tex] is not bounded above, I claim that [tex]S=(a, \infty)[/math] or [tex]S=[a, \infty)[/tex].
Case 1: Suppose [tex]a,x \in S[/tex] such that [tex]a \neq x[/tex]. Since [tex]a=g.l.b.(S)[/tex], [tex]a<x[/tex]. Now since [tex]S[/tex] is unbounded, there is some [tex]s \in S[/tex] such that [tex]s>x[/tex]. but since [tex]a,s,x \in S[/tex], by previously proved lemma, [tex][a,x] \in S[/tex] and [tex][x,s] \in S[/tex]. Henc e [tex]S=[a, \infty)[/tex].
Case 2: Suppose [tex]a \notin S[/tex] and suppose [tex]x \in S[/tex], [tex]x>a[/tex].

Now I am not sure how to move on to say that everything approaching a is in S but a is not in S.
 

1. What does it mean for a subset of real numbers to be bounded below?

Being bounded below means that there exists a real number, called the lower bound, that is less than or equal to all the numbers in the subset. In other words, the subset has a finite or infinite minimum value.

2. How can I determine if a subset of real numbers is bounded below?

To determine if a subset of real numbers is bounded below, you can compare the numbers in the subset and see if there is a minimum value. Alternatively, you can use the concept of infimum, which is the greatest lower bound of a subset, to determine if a subset is bounded below. If the infimum exists and is a real number, then the subset is bounded below.

3. What is the difference between bounded below and bounded above?

A subset of real numbers is bounded below if it has a finite or infinite minimum value, while a subset is bounded above if it has a finite or infinite maximum value. In other words, a subset that is bounded below has a lower bound, while a subset that is bounded above has an upper bound.

4. Can a subset of real numbers be both bounded below and unbounded above?

Yes, a subset of real numbers can be both bounded below and unbounded above. This means that there exists a minimum value for the subset, but there is no maximum value. For example, the set of all negative real numbers is bounded below by 0, but it is unbounded above because there is no maximum negative number.

5. Is a subset of real numbers bounded below and unbounded above always infinite?

No, a subset of real numbers can be bounded below and unbounded above and still be a finite set. For example, the set of real numbers between 0 and 1, including 0 and 1, is bounded below by 0 but is unbounded above because there is no maximum value. However, this set is finite because it only contains a finite number of values between 0 and 1.

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