Homework Statement: [/B]Let a = sup S. Show that there is a sequence x1, x2, ... ∈ S such that xn converges to a.
Homework Equations: [/B]I know the definition of a supremum and convergence but how do I utilize these together?
The Attempt at a Solution:[/B] Given a = sup S. We know that a = sup S if: 1) a ∈ S and a is called an upper bound, and 2) if b is also an upper bound, then b ≥ a. Since a = sup S, given ε>0 and the xn ∈ S, we know that a - ε < xn ≤ a since a is a least upper bound. This means that since xn ∈ S and a = sup S, that xn can never exceed the value of a, given as sup S.
** I am stuck, any help is beneficial.