- #1
CAF123
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Homework Statement
Suppose that E is contained in ##\mathbb{R}## is a nonempty bounded set and that ##\sup E## is not in E. Prove that there exists a strictly increasing sequence ##\left\{x_n\right\}## that converges to ##\sup E## such that ##x_n \in E## for all n in ##\mathbb{N}##.
Homework Equations
More like Relevant theorems:
Bolzano-Weirstrass,
Completeness,
Monotone convergence.
The Attempt at a Solution
E is contained in R and nonempty so by Completeness, sup E exists. E is bounded so -a < e < a for all e in E and let supE = a which is not in E. I suppose this might be the essence of the proof, but how exactly do we know such a sequence in E exists? If there existed a function ##f: \mathbb{N} \rightarrow \mathbb{R}## with all the ##x_n## in E, then provided I can show such a sequence is strictly increasing (and that it exists) then the rest of the problem becomes trivial. I thought about using nested intervals but I didn't get very far. Any ideas?
Many thanks.