I took $\sup A=\alpha$ and $\sup A=\beta$ I have to show that $\sup(A+B) = \sup A+\sup B$. I have showed that $\sup(A+B)$ exists first and also that $\alpha$ and $\beta$ is upper bound for set $A+B$. I just need to show that it is the least.
Claim: $\alpha + \beta$ is least upper bound
So I took $\gamma$ < $\alpha$ + $\beta$ where $\gamma$ is least upper bound. I am aiming to show that $\gamma=\alpha + \beta$
So I write $a_1+b_1 > \gamma - \epsilon$ for all epsilons, $a_1$ belongs to $A_1$ and so
$a_1+b_1 > \gamma - \epsilon$
Also $a_1 +b_1 \leq \alpha + \beta$
So i get $\gamma - \epsilon \leq a_1 +b_1 \leq \alpha + \beta $ for all e[silons
So $\gamma -(\alpha + \beta) < \epsilon$ for all $\epsilon >0$ and so I am done
IS THIS OKAY?
Thanks for help