MathproveLetEbeanon-emptysetofpositiverealnumbersanddefineS(E)=∑(k,i=1)xi:k∈Nand{xi(k,i=1)⊆E}(Thus,S(E)isthesetofallsumsof¯nitenumbersofdistinctelementsinE.ThatmeansS(E)consistsallsumsofthefor

Question

Math prove
Let E be a non-empty set of positive real numbers and define
S(E) = ∑(k,i=1)xi :k ∈ N and {xi(k,i=1) ⊆ E}
(Thus,S(E) is the set of all sums of ¯nite numbers of distinct elements in E.That means S(E) consists all sums of the form x1+x2+.+xk where k can be any natural number and the xi's
are k different elements in E.) If S(E) is unbounded,then we write ∑(x∈E)x=∞.Prove that if S(E) is bounded,then E must be countable.(In this case,we define ∑(x∈E)x = sup(S(E)) and we have that ∑(x∈E) is finite.

Answer 1

S(E)有限,而∑(x∈E)x是单调增加的,随k的增加而增加.那么∑(x∈E)x一定存在上确界,否则S(E)无限.那么假设k=m时达到最大值,则e中应有m个元素.
原理：单调有界数列必有极限.