Triangle inequality for suprema?

Question

Let $x \in A$ and $y \in B$, where $A, B \subseteq \mathbb C$. Should then \begin{equation} \sup_{x\in A, y \in B} \lvert x + y \rvert \leq \sup_{x\in A} |x| + \sup_{y\in B} |y|? \end{equation}

I'm sruggling with this idea a bit, since it's been a few years since I took an analysis class and I spent the entire last school year studying math teacher "stuff", which only included high school level math at most.

This seems trivial, if I'm allowed to use the triangle inequality for complex moduli, but there is always some catch that I miss with these. I know that $s_A = \sup_{x\in A} |x| \geq |x|$ for all $x \in A$ and a similar claim holds for all $y \in B$: $s_B = \sup_{y\in B} |y| \geq |y|$ for all $y \in B$. With the lower bound, $s_C = \sup_{x\in A, y \in B} \lvert x + y \rvert \geq |x + y|$ for all $x \in A$ and $y \in B$.

Since the triangle inequality holds for every $x, y \in \mathbb C$, it should then hold for arbitrary subsets of the complex plane, and especially for the suprema of the moduli. I am surely missing something here, but what is it? How can I make this more rigorous?

Answer 1

If $x \in A$ and $y \in B$ then $|x+y| \leq |x|+|y| \leq s_A+s_B$. Taking sup over $x$ and $y$ we see that $\sup_{x \in A, y \in B} |x+y| \leq s_A+s_B$.

Answer 2

1) $|x+y| \le |x|+|y|$;

$\sup |x+y|\le \sup (|x|+|y|)$;

2) $|x|+|y| \le |x|+\sup |y|$;

$\sup (|x|+|y|) \le \sup (|x|+\sup |y|) =\sup |x|+\sup |y|$.