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 Sep 20 accepted Why is it that $\left|b_n - b \right| < \frac{\left|b \right|}{2} \Rightarrow \left| b_n \right| > \frac{\left|b \right|}{2}$? Sep 20 comment Why is it that $\left|b_n - b \right| < \frac{\left|b \right|}{2} \Rightarrow \left| b_n \right| > \frac{\left|b \right|}{2}$? Thanks to all of you for your help! Part of what I overlooked is that $|b_n - b| = |b - b_n|$... I think I was trying to do this, but I kept getting $\frac{3|b|}{2}$ or something... Anyway, I think it makes sense to me finally :) Sep 20 asked Why is it that $\left|b_n - b \right| < \frac{\left|b \right|}{2} \Rightarrow \left| b_n \right| > \frac{\left|b \right|}{2}$? Sep 7 accepted What is the explanation for the elements of this set? Sep 7 revised What is the explanation for the elements of this set? edited body Sep 7 comment What is the explanation for the elements of this set? by the way I hope that $f(A)$ is the correct notation for the range of the function. Sep 7 comment What is the explanation for the elements of this set? thank you, that definitely helps a lot. I still have to ask: in order to "note that $B$ is not in the range of the function" we then need to consider the range of the function to be $\{ \{ a\}, \{ a,c\} \{ a,b,c\}\}$ as I wrote in the question? And that's why $B = \{b\} \notin f(A)$? Is that correct? In other words $B$ is a set, but the important observation is that it is not an element of the range of the function? Sep 7 comment What is the explanation for the elements of this set? Sorry, those weren't actually my solutions, I've updated the question to hopefully make that more clear... Sep 7 revised What is the explanation for the elements of this set? added 103 characters in body Sep 7 asked What is the explanation for the elements of this set? Aug 21 awarded Electorate Aug 20 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? @kahen: sorry, but I was having a little difficulty understanding the right to left direction of the statement you wrote. The way I was interpreting it, I would have thouhgt that if $p = 4$, $a=4$, $b=2$ so $ab = 8$, then $p$ divides $ab$ as well as $a$... But is part of the requirement that this is true for every $ab$? So if $p=4$ it divides $ab=12$ yet $4$ divides neither $a=6$ nor $b=2$ and therefore $p=4$ is not prime? Is that correct? Aug 20 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? thank you rbojohn Aug 20 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? thank you for this Aug 20 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? thank you very much Aug 20 accepted What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? Aug 19 revised What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? added 51 characters in body Aug 19 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? @Bill: for context, this is me trying to go through the book Understanding Analysis by Stephen Abbott. Unfortunately I do not know any number theory or abstract algebra... Aug 19 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? @kahen and John, thank you for your comments! Aug 19 asked What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers?