This question already has an answer here:
In the answer to this post Ittay Weiss wrote that, "...a sum of infinitely many positive elements can be bounded only if there are countably many elements." Though I asked about a rigorous proof of this fact and a reply was given, as can be easily seen, I still have some doubts which I need to clarify. I also didn't understand how Weiss's comment gives a proof of my question and so I am asking it here again.
It seems from the phrasing of the quote that there are other types of sum of positive elements for which the number of elements isn't countably infinite. What are some examples of sums of this type? To be precise, if sum of positive elements for which the number of elements isn't countably infinite then should it be necessarily uncountable?
If so, then how do we define a "sum" of uncountable number of positive quantities?
What is meant when we say that the "sum" is unbounded?
Why is a sum of infinitely many positive elements can be bounded only if there are countably many elements?
Any help will be appreciated.