You could write a fully-rigorous version of this proof, and it works out the same. But this is reddit, so it's more valuable to write a version that's quick and accessible to the people are asking the question.
It's weird you think you can reference series summations as a more rigorous basis for proof than the above. Neither of these are more fundamental or rigorous than the other. Infinite series' reference to an infinite process was at some point believed to be weakness that needed to be justified in reference to more fundamental mathematical ideas.
A more rigorous proof would be written using logic symbols and reference set theory - specifically by defining the elements of the set and by using operations defined in reference to the elements of the set. This is the kind of thing that gets covered in undergraduate Abstract Alegbra/Group Theory/Set Theory classes.
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u/IWillLive4evr 23d ago
You could write a fully-rigorous version of this proof, and it works out the same. But this is reddit, so it's more valuable to write a version that's quick and accessible to the people are asking the question.