x is an upper bound for the set Y. Prove that x = sup(Y) (least upper bound) if and only if for every e > 0, there is some y in Y (depending on e) such that x < y + e
(e in this case is every positive real number)
The Attempt at a Solution
since for every y in Y, y <= x. hence, x - y >= 0 for each y in Y. Thus, there corresponds some positive real number for each y in Y, which can be added to y to make it greater than x. x < y + e.
Does this make sense? sorry for the poor notation. Thank you.