# Sup and inf of a set of limit points

1. Oct 8, 2012

### Yankees24

1. The problem statement, all variables and given/known data

I have to prove that the supremum and infimum of a set of limit points of a a sequence {an} are themselves limit points.

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Oct 8, 2012

### LCKurtz

The general idea is, there are limit points close to the sup of the limit points. And there are points in the given set that are close to those limit points. So there are points in the set close to the sup of the limit points. You just have to write it carefully with appropriate inequalities.

3. Oct 8, 2012

### Yankees24

Great! Could you possibly give me an idea of where to begin with the careful proof? This is usually where I struggle. Thank you!

4. Oct 8, 2012

### LCKurtz

If you call original set of points $S$ and the sup of the limit points $s$ and you want to show $s$ is a limit point of $S$ you would start with the definition for $s$ to be a limit point of $S$. That is what you have to prove. And you have already neglected to mention what $S$ is a set of e.g., the real numbers.

5. Oct 8, 2012

### Yankees24

Ok thanks I will see how it goes. And yes I meant to say a sequence of nonnegative real numbers.