The way I look at this is that objects like 12X0+12X1 live in a function space like X→R≥0, specifically the subspace of that where the functions f are integrable with respect to counting measure on X and ∑x∈Xf(x)=1. In other words, objects like f1:=12X0+12X1 are probability mass functions (pmf). f1(X0) is 12, and f1(X1) is 12, and f1 of anything else is 0. When we write what looks like an infinite series λ1f1+λ2f2+⋯, what this really means is that we’re defining a new f by pointwise infinite summation: f(x):=∑∞i=1λifi(x). So only each collection of terms that contains a given Xk needs to form a convergent series in order for this new f to be well-defined. And for it to equal another f′, the convergent sums only need to be equal pointwise (for each Xk, f(Xk)=f′(Xk)). In Paul’s proof above, the only Xk for which the collection of terms containing it is even infinite is X0. That’s the reason he’s “just calculating” that one sum.
The way I look at this is that objects like 12X0+12X1 live in a function space like X→R≥0, specifically the subspace of that where the functions f are integrable with respect to counting measure on X and ∑x∈Xf(x)=1. In other words, objects like f1:=12X0+12X1 are probability mass functions (pmf). f1(X0) is 12, and f1(X1) is 12, and f1 of anything else is 0. When we write what looks like an infinite series λ1f1+λ2f2+⋯, what this really means is that we’re defining a new f by pointwise infinite summation: f(x):=∑∞i=1λifi(x). So only each collection of terms that contains a given Xk needs to form a convergent series in order for this new f to be well-defined. And for it to equal another f′, the convergent sums only need to be equal pointwise (for each Xk, f(Xk)=f′(Xk)). In Paul’s proof above, the only Xk for which the collection of terms containing it is even infinite is X0. That’s the reason he’s “just calculating” that one sum.