Let , I take this to be your question.
.
For notation:
but that means that (WHY?)
Now suppose that and find a contradiction to prove that .
I have an assignment quesiton :
" let A_i be a subset of the reals and i is element of I = (1,...,n)
now let A = UNION (i is element of I) A_i
show that sup(A) = sup_iEI(sup(A))"
I hope that is understandable...
so taking the limit of both sides I have limA = lim U A_i, I'm not sure what the limit of the union of a set is. Am I on the right track?