Proving Infinitely Many Segments in an Opened Set

[sutupidmath]

Hi,
i was just wondering how to prove that from every opened set we can create infinitely many segments within it. My point is this, if we have a scaled line, and if we choose an interval, let's say (a,b), then how can i prove that i can construct infinitely many segments whithin the given interval (a,b)?

I hope u got my point, couse i have been a little ambiguous.

 

[morphism]

What is a segment?

 

[sutupidmath]

Well, a segment is the set of all points,that lay on a line, between two fixed points, including these two,boundary, points. [a,b].
Something like this, right?

 

[ObsessiveMathsFreak]

Well, it's a subtle point, but you can't actually "construct" infinitely many segments. Rather what the question is asking you to prove is that there is no upper limit on the number of segments you can construct.

You would prove there is no upper limit as follows. First prove that you can divide any given line segment into finitely more line segments(two will do). Then assume there is an upper bound N, on the number of segments (a,b) can be divided into. But every one of those line segments can be divided up into more by the first part of the proof, and now (a,b) has been divided up inot more than N segments. So the assumption of an upper bound on the number of segments is false, and there is no upper limit on how many times you can segment an interval.