Expert: Paul Klarreich - 8/14/2011

Question
QUESTION: Let A be a nonempty subset of R that is bounded above.
(1) Set
D := {2a | a ∈ A}.
Is it necessarily true that sup D = 2 sup A? Either prove or provide a counterexample.
(2) Set
S :={a2 |a∈A}.
Is it necessarily true that sup S = (sup A)2 ? Either prove or provide a counterexample.

ANSWER: Questioner:Michael
Country:Australian Capital Territory, Australia
Category:Calculus
Private:No
Subject:Supremum - The Completeness Axiom

Question:

Let A be a nonempty subset of R that is bounded above.
(1) Set  D := {2a | a ∈ A}.
Is it necessarily true that sup D = 2 sup A? Either prove or provide a counterexample.
(2) Set  S :={a2 |a∈A}.
Is it necessarily true that sup S = (sup A)2 ? Either prove or provide a counterexample.
...........................................
Call S = sup A.

Suppose  x in D.  Then x <= S and so  2x <= 2S.  
So 2S is an upper bound of D.

Now suppose  Y  is any other upper bound of D.

Then  2x <= Y, for all x, and therefore:

x <= Y/2.

So Y/2 is an upper bound of A and therefore Y/2 >= S.

So Y >= 2S, which means .....


......................

Your second example has a typo, I think.  Fix it and resubmit, if you have not figured it out already.


---------- FOLLOW-UP ----------

QUESTION: Apologies about the typo. a and (sup A) should be squared:

(2) Set  S :={a^2 |a∈A}.
Is it necessarily true that sup S = (sup A)^2 ? Either prove or provide a counterexample.

Answer
Questioner:   Michael
Private:Yes   <<<<<<<<<<<<<<<<<<<<<< CHANGED TO PUBLIC.

QUESTION: Let A be a nonempty subset of R that is bounded above.
(1) Set
D := {2a | a ∈ A}.
Is it necessarily true that sup D = 2 sup A? Either prove or provide a counterexample.
(2) Set
S :={a2 |a∈A}.
Is it necessarily true that sup S = (sup A)2 ? Either prove or provide a counterexample.

...........................................
Call S = sup A.

Suppose  x in D.  Then x <= S and so  2x <= 2S.  
So 2S is an upper bound of D.

Now suppose  Y  is any other upper bound of D.

Then  2x <= Y, for all x, and therefore:

x <= Y/2.

So Y/2 is an upper bound of A and therefore Y/2 >= S.

So Y >= 2S, which means .....

......................

Your second example has a typo, I think.  Fix it and resubmit, if you have not figured it out already.

---------- FOLLOW-UP ----------

QUESTION: Apologies about the typo. a and (sup A) should be squared:

(2) Set  S :={a^2 |a∈A}.
Is it necessarily true that sup S = (sup A)^2 ? Either prove or provide a counterexample.


Ah! This is different.  Try  A = {x |  -3 <= x <= -2 }