Topology - Limit Points

[Aryth]

Suppose \(\displaystyle M\) is a nonempty point set of \(\displaystyle \mathbb{R}\) and \(\displaystyle b\) is a point to the left of every point in \(\displaystyle M\). If \(\displaystyle M\) has no leftmost point, prove that \(\displaystyle M\) has a limit point.

I have thought about this for some time, and I cannot figure out what \(\displaystyle b\) has to do with it... I was not given enough information prior to this to even begin to understand what this means. If someone could help me along that would be awesome.

 

[daon2]

Suppose \(\displaystyle M\) is a nonempty point set of \(\displaystyle \mathbb{R}\) and \(\displaystyle b\) is a point to the left of every point in \(\displaystyle M\). If \(\displaystyle M\) has no leftmost point, prove that \(\displaystyle M\) has a limit point.

I have thought about this for some time, and I cannot figure out what \(\displaystyle b\) has to do with it... I was not given enough information prior to this to even begin to understand what this means. If someone could help me along that would be awesome.

Do you know what greatest lower bounds/infimums are?

What did you cover prior to being assigned this question?

 

[pka]

Suppose \(\displaystyle M\) is a nonempty point set of \(\displaystyle \mathbb{R}\) and \(\displaystyle b\) is a point to the left of every point in \(\displaystyle M\). If \(\displaystyle M\) has no leftmost point, prove that \(\displaystyle M\) has a limit point.

It is impossible to give an exact answer without knowing your notes.
Here is the idea. \(\displaystyle b\) is a lower bound of \(\displaystyle M\). So the set of lower bounds of \(\displaystyle M\) is nonempty and bounded above by the fact that \(\displaystyle M\) is nonempty, So the set of lower bounds of \(\displaystyle M\) has a least upper bound. Call it \(\displaystyle \beta\). How do we know that \(\displaystyle \beta\notin M~?\) So prove that \(\displaystyle \beta\) is a limit point of \(\displaystyle M\).