Connected Subset of Real Numbers: Bounded Below, Unbounded Above

[tarheelborn]

If $$S$$ is a connected subset of $$\mathbb{R}$$ and $$S$$ is bounded below, but not above, then either $$S=[a, \infty)$$ or $$S=(a, \infty)$$ for some $$a \in \math{R}$$.

 

[Office_Shredder]

What have you tried so far? Do you know what connected subsets of R look like?

 

[tarheelborn]

Yes I am familiar with the types of connected subsets of $$\mathbb{R}$$. I know that if $$a<b \in S$$ then $$[a,b] \subseteq S$$. I think I need to find some point in $$[a,b]$$ and show that $$a$$ is the least greatest lower bound of that interval whether or not $$a$$ is included in the interval. But I am not sure how to start.

 

[tarheelborn]

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!

 

[Office_Shredder]

Yes I am familiar with the types of connected subsets of $$\mathbb{R}$$. I know that if $$a<b \in S$$ then $$[a,b] \subseteq S$$.

More specifically than just this, the only connected subsets of R are intervals

 

[tarheelborn]

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 $$[a, \infty)$$ or $$(a, \infty)$$ 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.

 

[tarheelborn]

I have gotten this far with the proof:


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