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The linked proof is in this post. It first assumed $$(1)\quad\quad |f(x_n)-f(y_n)|\geq \varepsilon_0\;\;\;\;n\in\mathbb{N}.$$ Then after finding convergent sub-subsequences, $\{x_{n_{k_j}}\}$ and $ \{y_{n_{k_j}}\}$, the proof then says it follows from (1) that $$(2) \quad\quad |f(x)-f(y)|\geq \varepsilon_0.$$

Questions:

  1. The (2), in the proof, seems to now claim $\forall x,y\in I$. Is this true?
  2. Additionally, how do the sub subsequences play a role in finishing the proof to find the contradiction? Was it used in showing that (2)? If this is the case, I don't know then.

Any help is appreciated as it is not obvious for me, thank you!

user10101
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  • You should first write down in a clear way all the steps you have made – blamethelag Oct 26 '21 at 20:00
  • steps? @blamethelag I am not sure I understand your question as the steps (and i guess the implicit ones) of the proof are in the linked post (Then I guess I am looking where the implicit things are hidden to then assume the second line). Would you clarify what you mean? – user10101 Oct 26 '21 at 20:07
  • In my opinion what you wrote on your post is hasty and confused. You should rephrase it in a more clear way so that you understand what is happening. – blamethelag Oct 26 '21 at 20:12
  • edited @blamethelag – user10101 Oct 26 '21 at 20:37
  • Oh my sorry. I meant you should do the proof from the begining to where you are stuck. Like perform the redaction, introduct the objects etc – blamethelag Oct 26 '21 at 20:39

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