The sum of an uncountable number of positive numbers

Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).

Proof: Let $\sum_{\alpha\in A}x_\alpha=M<\infty$. Consider $S_n=\{\alpha\in A \mid x_\alpha>1/n\}$.

Then $M\geq\sum_{\alpha\in S_n}x_\alpha>\sum_{\alpha\in S_n}1/n=\frac{N}{n}$, where $N\in\mathbb{N}\cup\{\infty\}$ is the number of elements in $S_n$.

Thus $S_n$ has at most $Mn$ elements.

Hence $\{\alpha\in A \mid x_\alpha>0\}=\bigcup_{n\in\mathbb{N}}S_n$ is countable as the countable union of finite sets. $\square$

First, is my proof correct? Second, are there more concise/elegant proofs?

Just so the question gets an answer: yes, your proof is correct and is one of several phrasings of the shortest proof that I know.

There is also one question directly relating to this question, that is, how to define the sum of uncountably many numbers (not necessarily positive numbers). The difficulty lies in the fact that there could not be any order of this summation, since there are uncountably many of terms. So, when we talk about the sum of $x_\alpha$, namely, $\sum_{\alpha\in A}x_\alpha$, we are actually saying the following,

For every countable subset of $I\subset A$ with arbitrarily given order, the sequence $(x_\alpha)_{\alpha\in I}$ should be convergent. In other words, the sequence $(x_\alpha)_{\alpha\in I}$ should be absolutely convergent.

A proper definition is given in Paul Halmos' book, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, as follows:

$x=\sum_{\alpha\in A}x_\alpha$ means that for any positive number $\varepsilon$ there is some finite set $I_0$ such that for any finite set (or more generally, countable set) $I\supset I_0$, we have $|x-\sum_{\alpha\in I}x_\alpha|<\varepsilon$.

Note that, the set $\{1,-\frac{1}{2},\frac{1}{3},\cdots,(-1)^{k-1}\frac{1}{k},\cdots, 0,\cdots\}$, where in the end there are uncountably many $0$'s is not convergent any more. But the sequence $\{1,-\frac{1}{2},\cdots\}$ is convergent in the standard sense.

Now we invoke Zorn's lemma, on all countable subsets $I$, with respect to which, the sequence $x_\alpha$ is absolutely convergent, with the inclusion as the order. Note that for any $I_1\subset I_2\subset I_3\subset \cdots$ a chain of countable subsets of $A$, the set $I^*=\bigcup_iI_i$ is also a countable subset of $A$ and by the definition, the sequence with index in $I^*$ is also absolutely convergent. By Zorn's lemma, there should exists a maximal countable subset $I_{max}$. This means that any number $x_\alpha$ with $\alpha\notin I_{max}$ should be $0$, otherwise we can construct another strictly larger countable subset on which the number sequence is absolutely convergent.

A more explicit formulation of the statement in the title is this. Let $A$ be some uncountable family of non-negative numbers. If $S$ is to be the "sum of $A$", then by any reasonable definition of "sum", surely $S$ must be greater than or equal to the sum of any finite sub-family of $A$. In that sense we can prove that any for any reasonable definition of "sum", the sum of $A$ must be infinite (unless all but countably many elements of $A$ are zero) by showing:

For any $M$, there exists a finite sub-family $B$ of $A$ such that the sum of $B$ is at least $M$.

Proof: Assume that $A^+$, the positive members of $A$, is uncountable (otherwise the theorem is obviously false). $A^+=\bigcup_n A_n$, where $A_n=\{a\in A | a \geq \frac 1 n\}$. Since the union of countably many finite sets is countable, one of the $A_n$ must be infinite. Grab as many elements as you need from that set to get a sum greater than $M$.

The question is not well-posed because the notion of an infinite sum $\sum_{\alpha\in A}x_\alpha$ over an uncountable collection has not been defined. The "infinite sums" familiar from analysis arise in the context of analyzing series defined by sequences indexed over $\mathbb{N}$, and the series is defined to be the limit of the partial sums. The only objects defined here are (1) finite sums, and (2) limits of sequences indexed by the natural numbers.

If the question is interpreted as asking whether, out of an uncountable collection $X$ of positive reals, one could always form a divergent series, the answer is affirmative, and moreover one can always choose a sequence $(u_n)$ in $X$ with a fixed positive lower bound $u_n>\epsilon>0$. This is because $X$ admits a countable cover consisting of sets $A_r$ of members of $X$ that are greater than $r$ for all rational $r$, and therefore one of the $A_r$ is uncountable and in particular infinite.

Note that over the hyperreals, one does have a notion of infinite sum, when the collection being summed over is hyperfinite (it is necessarily uncountable).