Taken from sec. 1.4.1 of the book by Mary Hart, titled: Guide to Analysis.
Let $A, B$ be two non-empty sets of real numbers with supremums $\alpha, \beta$ respectively, and let the sets $A + B$ and $AB$ be defined by :
$A + B = {a + b: a\in A, b\in B}$,
$AB= {ab:a\in A, b\in B}$.
Show that $\alpha+ \beta$ is the supremum of A + B.
My attempt :
If a non-empty set of reals $A$ is having supermum $\alpha$, & other set $B$ is the supremum $\beta$; then two cases are possible for each supermum:
(i) supremum is in set, and then it equals the maximum; as say $(0.1,0.2]$,
(ii) supremum is not in set, & maximum does not exist; as say $(0.1, 0.2)$.
In case of combination of such two sets possibilities arise based on larger value. However, its inclusion, or exclusion (i.e. $],)$) does not affect supermum, as it always exists.
Say, set $A$ is $(a,b)$ & set $B$ is $(c,d)$, then the sum has upper limit decided by the relative magnitudes of $a,b,c,d$. So, if $a\gt b \gt c \gt d$; then supermum is $b$, & no maximum.
But, it is not clear how to prove the question.
P.S. There are few other questions from the same book, & request link to their answers; as could not find (even, could not find answer to this!).
Till now, I resisted stating these 'other' questions, as wanted to post seperate questions (to prevent clutter), but if links were provided then no need.
These questions are stated below:
- Give an example to show that AB need not have a supremum.
Prove also that even if AB has a supremum, this supremum need not be equal to $\alpha \beta$.
Show that if $A$ be set of positive reals with supremum $\alpha$, & let $Y = {x^2 : x\in X}$; then $\alpha^2$ is supremum of Y.