Lower bounds for two sets summed are still a lower bound of the sets summed

Question

Let $A,B \subset \mathbb{R}$ be bounded below. Let $7$ be a lower bound for $A$ and $12$ be a lower bound for $B$, show that $19$ is a lower bound for $A+B$

$\forall x \in A, 7 \le x$ and $\forall y \in B, 12 \le y $

now, $7 \le Inf(A), 12 \le Inf(B)$

$7 + Inf(B) \le Inf(A) + Inf(B) \,\, \& \,\, 12 + Inf(A) \le Inf(A) + Inf(B)$

$ 7+12 \le Inf(A) + Inf(B)$

Let $x+y \in A+B$, $x+y \le x+y \implies x \le x+y - Inf(B) \implies Inf(B) \le x+y - Inf(A) $

$\implies Inf(A) + Inf(B) \le x+y \implies 7+12=19 \le x+y$ And so 19 is a lower bound

I feel like there may be a more simplified way of doing this but this looks correct to me

Answer 1

More generally, if $\inf A = a$ and $\inf B = b$, then for any $x\in A+B$, $x=a'+b'$ where $a'\in A$, $b'\in B$. Hence $$x=a'+b'\geqslant a+b, $$ so that $a+b$ is a lower bound for $A+B$. It follows that $\inf A+B \geqslant \inf A + \inf B$.