(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let A and B be subsets of R (real numbers). The vector sum of two sets A and B is written as A+B and is defined to be:

A+B = {a+b : a in A, b in B}

Prove that for all bounded nonempty sets A and B, sup (A+B) = sup A + sup B

3. The attempt at a solution

let A* = sup A, B*=sup B and C*=sup(A+B)

(1) we first prove that C*<= A* + B*

- I understand the proof for this part

(2) Next, we prove that A* + B*<= C*

- This is MY proof:

since C* is the sup (A+B), then for any a+b in set (A+B):

a+b <= C*

Thus, C* is an upper bound for a+b, for any a in A and b in B

This implies that C* is also the upper bound for the sum of the highest possible a in A (namely, A*) and the highest possible b in B (namely, B*). Therefore:

A*+B* <=C*

In conclusion, by (1) and (2):

A*+B*=C*

My question, what is wrong with the logic in proof (2)? The solution has something different but I want to check if my thought process is also correct.

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# Homework Help: Proof of sup(A+B) = sup A + sup B | is my thought process correct?

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