Question

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' ≠∅

Answer 1

@eliesaab @zzr0ck3r

Answer 2

Hey this should be really easy. Simply consider the set \((upper~ bound ~of~ S, ~~\infty)\).

Is it a subset of \(S\) ?

Answer 3

Another way of seeing it if S' is empty, then S=R and R is not bounded

Answer 4

That argument is elegant!

Answer 5

Thanks Guys

Answer 6

Both the sets [0,1] and (0,1) are infinite bounded subsets of R.

The question is surely fairly trivial since R is unbounded (1)

So partitioning R into subsets S and S' where S' is the null set and S is R is contradicts (1)

How can i conclude this?

Answer 7

@eliesaab gave you an one line proof. It is the complete proof..

Answer 8

Here is the same proof with more words :

(proof by contradiction)
If S' is empty, then S=R and since R is not bounded, S is also not bounded.
But it is given that S is bounded. So, S' cannot be empty.

Answer 9

Thanks for your help guys