Proof Bounded sets
I'm stuck on how to answer this proof and I was wondering if anyone could help!
Prove that a subset S of R is bounded if and only if there
exists a real number H such that |x| ≤ H for all x ∈ S.
Thanks.
Prove that a subset S of R is bounded if and only if there
exists a real number H such that |x| ≤ H for all x ∈ S.
Thanks.
The way you approach this would depend on exactly what definition of 'bounded' you've been given. I'll assume you're using the one that says the set is bounded above and bounded below.
The (<=) direction should be fairly straightforward by picking appropriate upper and lower bounds.
For the (=>) direction, try and first do some stuff with max(__,__) and min(__,__) to find an upper bound that is positive and a lower bound that is negative. From there, you can then play with max(__,__) and min(__,__) again to find H such that H is an upper bound and -H is a lower bound.
The (<=) direction should be fairly straightforward by picking appropriate upper and lower bounds.
For the (=>) direction, try and first do some stuff with max(__,__) and min(__,__) to find an upper bound that is positive and a lower bound that is negative. From there, you can then play with max(__,__) and min(__,__) again to find H such that H is an upper bound and -H is a lower bound.
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