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## Homework Statement

: [/B]Let a = sup S. Show that there is a sequence x_{1}, x

_{2}, ... ∈ S such that x

_{n}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 x_{n}∈ S, we know that a - ε < x

_{n}≤ a since a is a least upper bound. This means that since x

_{n}∈ S and a = sup S, that x

_{n}can never exceed the value of a, given as sup S.

** I am stuck, any help is beneficial.