Let A be a nonempty subset of R that is bounded above. Show that there is a sequence of elements of A that converges to sup A.
Can we say that since A is bounded above, so is ( ), then ( ) is nondecreasing since it's bounded above, and so sup{ } exist.
No, you can't say anything at all about until after you prove that it exists at which point you are done.
What you can do is assert that, with a= sup A, for any positive integer n, there exist a point of the set A in the interval (a, a+1/n). (Why?) Define to be that number.