How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$

If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?

• The same way you show anything about a supremum: show that the right side is an upper bound of A+B, and then show that any other upper bound is larger. – Nate Eldredge Sep 13 '10 at 17:25
• I have a question, that , if we consider $A = (-1)^n$ & $B= 1/n$ . In this will $SUP A+B$ =$SUP A + SUP B$? – Rkb May 30 at 12:18

Show that $\sup(A+B)$ is less than or equal to $\sup(A)+\sup(B)$ by showing that the latter is an upper bound for $A+B$. Then show that $\sup(A)+\sup(B)$ is less than or equal to $\sup(A+B)$ by showing that $\sup(A+B)$ is an upper bound for $A+\sup(B)$ and that $\sup(A+\sup(B)) = \sup(A)+\sup(B)$.

Set $x=\sup A$, $y=\sup B$. Given $a\in A, b\in B$ we have $a\le x, b\le y$ so $a+b\le x+y$ so it's an upper bound. Now take $\varepsilon > 0$ and find $a,b$ such that $a>x-\varepsilon/2, b>y-\varepsilon/2$ and you have $a+b > x+y - \varepsilon$. That suffices (since it means that every potential "smaller upper bound" $x+y-\varepsilon$ is not really an upper bound).

The following proofs are based on Question 2 here. The nub is to prove two inequalities: $$\sup(A+B) \geq \sup A+ \sup B \tag{1}$$

$$\sup(A + B) \leq \sup A + \sup B \tag{2}$$

Proof of $$(2)$$:

By definition of $$A + B$$ and $$\sup(A + B)$$, for all $$a \in A$$ and $$b \in B$$,

$$\color{seagreen}{a + b} \color{seagreen}{\leq \sup (A + B)}.$$ Subtract $$b$$ from both sides:

$$a = \color{seagreen}{a + b} - b \color{seagreen}{\leq \sup (A + B)} - b.$$

Hence, if we fix $$b \in B$$, then $$\color{seagreen}{\sup (A + B)} - b$$ is an upper bound for $$\color{seagreen}{A + B} - B = A.$$

And so by definition of $$\sup A$$, for every $$b \in B$$, $$\sup A \leq \sup (A+ B) − b.$$ Rearrange the previous inequality: $$\color{magenta}{b} \leq \sup(A +B) − \sup A$$ for all $$b \in B$$.

Hence, $$\sup(A +B) − \sup A$$ is an upper bound for any $$\color{magenta}{b}$$.

By the definition of supremum, the previous inequality means: $$\color{magenta}{\sup B} \leq \sup(A + B) − \sup A \iff \sup A + \sup B \leq \sup(A + B).$$

Proof of $$(1)$$:

Since $$\sup A$$ is an upper bound for $$A$$, $$a \leq \sup A$$ for all $$a \in A$$. Then $$b \leq \sup B$$ for all $$b \in B$$. Hence $$a + b \leq \sup A + \sup B$$ for all $$x \in A$$ and $$y \in B$$. Hence, $$\sup A + \sup B$$ is an upper bound for $$A + B$$. Hence, by definition of supremum, $$\sup A + \sup B \geq \sup(A + B)$$.

• Please correct me if I'm wrong, but I'm not entirely convinced by the latter half. $\sup (A+B)$ does not have to be part of the set $A+B$, right? So I'm not sure how your last statement holds - $\sup A + \sup B$ does not have to be $\ge A+B$ – shreyasgm Oct 4 '17 at 1:22
• @shreyasgm I may be wrong. How can I correct this pls? – Analysis Mar 21 '18 at 6:07
• @TuckerRapu I improved your formatting, I hope that is alright. As an aside, why did you include the link at the top? – Brahadeesh Aug 13 '18 at 18:28
• @Brahadeesh No problem! That was thes ource. – Analysis Jan 12 at 3:26
• @TuckerRapu Oh, I see. Nice answer, btw. :) – Brahadeesh Jan 12 at 4:08

Another proof for $\sup(A + B) \geq \sup A + \sup B$.

Proof when $\sup A + \sup B$ is finite:

Posit $e > 0.$ Then there exists $a \in A$ and $b \in B$ such that $x > \sup A − \frac{e}{2}$ and $y > \sup B − \frac{e}{2}$. Then $a + b \in A + B$. Ergo, $$\color{seagreen}{\sup(A + B)} \geq a + b \color{seagreen}{> \sup A + \sup B - e} \implies \color{seagreen}{ \sup(A + B) > \sup A + \sup B - e }.$$ Since $e > 0$ is arbitrary, $\sup(A + B) \geq \sup A + \sup B$

Proof when $\sup A + \sup B$ is infinite:

Since the sets here are nonempty, the suprema here are not equal to $-\infty$, so we're not in danger of encountering the undefined sum $-\infty +\infty$. If $\sup A + \sup B = + \infty$, then at least one of the suprema, say $\sup B$, equals $+\infty$. Select some $a_0 \in A$. Then $$\sup(A+B) \geq \sup(a_0 + B) = a_0 + \sup B = + \infty,$$ so $\sup(A+B) \geq \sup(A) + \sup(B)$ holds in this case. (Source)

$$\forall ~~a\in A,~~b\in B, a\le \sup A~~ \text{and }~~b\le \sup B \implies a+b \le a\le \sup A +\sup B$$

hence $$\sup A +B\le \sup A +\sup B$$ Now we know there exist $a_n\in A$ and $b_n\in B$ such that $$\sup A=\lim_{n\to\infty} a_n$$ and $$\sup B =\lim_{n\to\infty}b_n$$

then $$\sup A+\sup B =\lim_{n\to\infty} a_n+\lim_{n\to\infty} b_n =\lim_{n\to\infty} a_n+b_n \le \sup A+B$$

whence $$\sup A+\sup B = \sup A+B$$