Math Is Fun Forum

Sheff

Member

Registered: 2016-09-27

Posts: 4

Limit Points of a Set

I am aware a set is Bounded if it has both upper and Lower bound and i know what a Limit point of a set is but how can i show that If S ⊂ R be a "bounded infinite set", then S' ≠∅

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Limit Points of a Set

By S', do you mean the set of all limit points of S?

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Limit Points of a Set

Okay, here is a proof.

Construct a sequence with elements in S such that all terms are different. This should be possible because S is infinite. Now, because this sequence is bounded, it must have a convergent subsequence which is non-constant.

Hence, it follows that S' is non-empty.

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.