Sup and inf of a set of limit points

[Yankees24]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I have been messing around with definitions but have not made any progress. Please help. Thank you

 

[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.

 

[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!

 

[LCKurtz]

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

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.

 

[Yankees24]

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