Suppose I wanted to prove that for some positive constants $d$ and $e$ $$d\sup_{x\in \textbf{R}^{+}}x + e\sup_{y\in \textbf{R}^{+}}y = +\infty$$
How would I do this? Obviously this an intuitive example, however, it doesn't feel correct to simply say something like:
Because both supremums equal infinity, multiplied by any positive coefficient is still infinity, and thus the sums are still infinity
Since saying it "equals" infinity is just notational shorthand to saying the supremum is unbounded above.
How do I properly argue about this formally?
Edit: For clarity, I'm not married to the specific example. I'm just concerned about how to generally argue about sums/products of unbounded supremums.
The precise answer to this question is that since $\sup x=\infty$, you cannot multiply it with a number and add it to yet something else, because $\infty$ is not a number, and "$=\infty$" is just a shorthand for a longer sentence about some quantity being unbounded above.
Now you can theoretically define some algebraic rules for $\infty$ and thus artificially make it a number, such as $k\cdot\infty=\infty$ when $k$ is a positive real number, $\infty +\infty = \infty$, etc. But then your question is void, the equality that you want to prove follows trivially from these rules that you decided to add.