${A\_n}$ is a sequence of increasing events. Then I want to prove that $P(\cup A\_n)$ is the least upper bound of the sequence $P(A\_n)$. I am able to prove that it is an upper bound, but stuck in proving that it is indeed "least" upper bound.

BTW, I don't want the usual proof of "continuity of probability" from the probability axioms where we break the union into disjoint sets.