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- Jun 22, 2012

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**Equivalent Statements to Compactness ... Stromberg, Theorem 3.43 ... ...**

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.43 on pages 105-106 ... ...

Theorem 3.43 and its proof read as follows:

At about the middle of the above proof by Stromberg we read the following:

" ... ...Next select \(\displaystyle x \in A\) such that \(\displaystyle \rho (x, z) \lt \epsilon / 2\) [\(\displaystyle A\) is dense] ... ... "

My question is as follows:

Can someone demonstrate rigorously how \(\displaystyle A\) is dense in \(\displaystyle X\) guarantees that we can select \(\displaystyle x \in A\) such that \(\displaystyle \rho (x, z) \lt \epsilon / 2\) ... ...

Stromberg defines dense in X as follows ... ...

A set \(\displaystyle A \subset X\) is dense in \(\displaystyle X\) if \(\displaystyle A^{ - } = X\).

Hope someone can help ...

Peter

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It may help MHB readers to have access to Stromberg's terminology associated with topological spaces ... so I am providing access to the main definitions ... as follows:

Hope that helps ...

Peter

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