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I remember some Real Analysis textbook advising the same as Carl Morris in this proof. Does anyone know which book(s) and page(s)?

As a note, if I were writing the proof, I would write the proof with all my bounds $η$ and then choose $η$ to make the conclusion match the arbitrary $ϵ$.

Stephen Abbott's proof in that link presciently picks bounds that all add up in the end neatly to $\epsilon$, but that aren't intuitive or obvious. Carl Morris's advice would make choosing bounds and the proof more intuitive and less fey!

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    The way a proof is written (e.g., in textbooks or papers) is not necessarily the way the author conceived them. I suspect most people do something similar to what Carl Morris suggests, but then just go back and plug in the choice of $\eta$ in terms of $\epsilon$, so that it seems like they "presciently" knew exactly what was needed to get things to "neatly sum to $\epsilon$." – angryavian Apr 22 '20 at 04:40
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    That's definitely what I do anyways. – Clement C. Apr 22 '20 at 04:45
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    Also I have seen that sometimes authors get fed up of this and instead write the final in terms of $(a+b+c+d+\dots+k) \epsilon$ in one of the proofs in later parts of the book. The key is that it does not matter at all. The reason why it is instructive to bring final inequality in terms of $\epsilon$ only is that often it may be difficult to check that $a, b, c, d$ etc are constants independent of $\epsilon$. – Paramanand Singh Apr 22 '20 at 04:54

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I lit upon A Basic Course in Real Analysis | Mathematical Association of America, written by Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.

I also wish more books would employ a technique for doing limit arguments that I first saw explicitly described in [Arthur] Mattuck’s Introduction to Analysis; he refers to it as the “$K-ε$ principle” and describes it thusly: “let’s once and for all agree that if you come out in the end with $2ε$ or $22ε$, that’s just as good as coming out with $ε$. If $ε$ is an arbitrary small number, then so is $2ε$. Therefore, if you can prove something is less than $2ε$, you have shown it can be made as small as desired.” This principle, though mathematically obvious, saves a lot of time and effort in doing limit theorems, and, I think, also helps clarify for the student what “$ε-$arguments” really mean.