- #1
dancergirlie
- 200
- 0
Homework Statement
Let A contained in R be a set of real numbers. For c in R define set cA as
cA: {x in R|x=ca for some a in A}
a. prove that if c is greater than or equal to zero, then cA is bounded above and sup(cA)=cSup(A).
b. prove that if c is less than zero, then cA is bounded below and that the inf(cA)=csup(A)
Homework Equations
The Attempt at a Solution
a. Assume A contained in R is bounded above and let c in R be greater than or equal to zero. Since A is bounded above, that means for all a in A, there exists a b in R so that:
a is less then or equal to b for all a in A.
Multiplying by c yields:
ca is less than or equal to cb, where ca is in cA and cb is in R.
Therefore there exists an element r in R so that ca is less than or equal r for all ca in CA.
Meaning, that cA is bounded above.
Since cA is bounded above, that means by the completeness axiom, that cA has a least upper bound, call it Sup(cA). Where for any epsilon greater than zero,
sup(cA) - epsilon is less than ca for all ca in cA.
Similarly, since A is bounded above, that means that A has a least upper bound, call it Sup(A). Where for any epsilon greater than zero,
sup(A)-epsilon is less than a
multiplying by c yields:
cSup(A) -epsilon is less than ca.
This is where I get stuck, I don't know how to show that cSup(A) is equal to Sup(A). Any help/hints would be appreciated!