Let $f$ and $g$ be bounded, real-valued functions defined on $[a,b]$. Define \begin{align*} A &= \{f(x_1)+g(x_2) : x_1,x_2\in[a,b]\},\\ B &= \{f(x)+g(x) : x\in[a,b]\}. \end{align*} How are inf$A$ and inf$B$ related? Prove your answer.
My attempted solution is as follows:
$\mbox{inf}A=\mbox{min}f+\mbox{min}g\geq\mbox{min}(f+g)=\mbox{inf}B$.
I'm not entirely convinced that this is correct, and even if it is, I'm unsure of how to prove it.
A rigourous proof :
You have $B\subseteq A$.
By definition, $\forall x\in A, x\geq \inf(A)$ and $\forall x\in B, x\geq \inf(B)$.
($\inf(A)$ and $\inf(B)$ exist because f and g are bounded on $[a,b]$)
Let $b\in B$.
Since $B\subset A$, $b\in A$.
Consequently, by the above definition, $b\geq \inf(A)$.
So now we know that $\forall b \in B, b\geq \inf(A)$.
So $\inf(A)\in\lbrace x\in\mathbb{R} |\forall b \in B, b\geq x\rbrace$.
Or, by definition, $\inf(B)=\max\lbrace x\in\mathbb{R} |\forall b \in B, b\geq x\rbrace$.
So by definition of the max function, $\inf(A)\leq\inf(B)$.