- #1

Eclair_de_XII

- 1,067

- 90

## Homework Statement

"If ##x=sup(S)##, show that for each ##\epsilon > 0##, there exists ##a∈S## such that ##x-\epsilon < a ≤ x##"

## Homework Equations

##x=sup(S)## would denote the least upper bound for ##S##

## The Attempt at a Solution

"First, we consider the case where ##x=sup(S)∈S##. Then ##x=max(S)##, and so there exists an ##a∈S##, namely ##a=x=max(S)##, such that ##x-\epsilon<a≤x##."

"Next we consider the case where ##x=sup(S)∉S##. Then because ##x## is the least upper bound for ##S##, the interval ##(x-\epsilon,x)## is guaranteed to be a non-empty subset of ##S##. Therefore, there exists ##a∈S## such that ##a∈(x-\epsilon,x)⊂(x-\epsilon,x]##."

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