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Continuity and Sequential Limits

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Respected Sir / Madam,

I need each and every step. please explain me in detail.

In page 529 Theorem III.(I need proof). Text Book:- Taylor & Menon

Theorem: - Prove that Any Bounded subset of R has a supermium and Infimum?
I need proof.

Thank you

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Continuity and Sequential Limits are investigated.

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Theorem III: Let S be the bounded and close point set, and let S be the function defined on S which is continuous at each point of S. then the values of f are bounded.

Proof: Let S be the bounded and close point set. In order two prove theorem, we assume contradiction. i.e. f is not bounded. Since f is not bounded, then there exits at least one point set which does not contain its finite value. Then by definition of continuous function there exits convergent sequence Pn which converges to P if and ...

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