Prove Set of Real Numbers Unbounded: Tips & Examples

[xlalcciax]

How to prove that set of real numbers is unbounded?

 

[micromass]

What is your definition of unbounded?

 

[reef]

How to prove that set of real numbers is unbounded?

It most likely involves showing that the cardinality of the real numbers exceeds ℵ ("aleph null").

 

[Fredrik]

It most likely involves showing that the cardinality of the real numbers exceeds ℵ ("aleph null").

That would be "uncountable". "Unbounded" has a bunch of different but equivalent definitions, like "is not a subset of some open ball".

 

[reef]

True. I figured that you would have to go about it the same type of way. Do you have to establish that a set without bound has no limit? Regardless, I'd be interested to see the proof when xlalcciax figures it out.

 

[Fredrik]

It's much easier to prove that ℝ is unbounded than to prove that ℝ is uncountable. Once you've written down a definition of "bounded" and thought about what it means, you're pretty much done.

 

[HallsofIvy]

I wish xlalcciax would get back to us with his definition of "bounded" so we would know in which direction to go.

 

[reef]

You could be really lazy and just say that the real numbers can be proven to be uncountable by means of Georg Cantors diagonalization process.. lol.