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How should I approach this? I understand visually that it makes sense, but I have no idea how to use math regarding supremum and infimum. In general, I am also struggling with these type of proofs because the worse thing is I have no idea where to start, how can I improve?

The following is the previous parts of the question, which may be useful: enter image description here

The main problem actually is I cannot substitute definition of supremum and infimum with definition of max and min, otherwise this problem is easy.

nabu1227
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  • For a formal proof you have to apply the formal definitions or previously proved statements. Actually, here you can combine (B) and (A), (A) applied to $f$ and $-g$. – Berci Jan 20 '20 at 22:34
  • @Berci, Just to verify what you mean is to try $\sup(f-g) \leq \sup{f}+\sup{(-g)}$? – nabu1227 Jan 20 '20 at 22:55
  • No, sorry, they might not be (easily) applicable here. – Berci Jan 21 '20 at 00:06

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Hint: Assume $f$ is not constant. Take two sequences $x_n$ and $y_n$ such that $f(x_n) \to \sup f$ and $f(y_n) \to \inf f$. Then $$f(x_n) - f(y_n) = |f(x_n) - f(y_n)| \le |g(x_n) - g(y_n)| \le\sup g-\inf g$$

Berci
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  • Thank you. How did you realize that sequences are appropriate tools to prove this? (I would like your thought process if possible). Also just to make sure, I could take the limit to get the required results? – nabu1227 Jan 21 '20 at 00:25
  • Also, can sequences still be created if f is merely bounded, i.e. it could be closed? (I had the impression of sequences related to open sets) – nabu1227 Jan 21 '20 at 00:31
  • Yes, just take the limit. Alternatively, we can directly use the $\varepsilon$-definition of supremum. Nevertheless, from there to sequences, just apply the def. to find an $x_n$ with $\varepsilon=\frac1n$. It's common in the context of $\inf, \sup$, so most of the times they just behave the same as $\min, \max$, which in effect realize their values. – Berci Jan 21 '20 at 00:32
  • Maybe it would be better if you choose different indices for the sequences, say $x_n$ and $y_m$. – jijijojo Jan 21 '20 at 00:42