Hello, I am working with Israel Gelfand's book "Algebra" and have another question about one of the exercises!
Fractions and are called neighbor fractions if their difference has numerator ±1, that is, ad–bc = ±1. Prove that
(a) in this case neither fraction can be simplified (that is, neither has any common factors in numerator and denominator);
(b) if and are neighbor fractions, then is between them and is a neighbor fraction for both and ; moreover,
(c) no fraction with positive integer and such that f < b+d is between and .
I understand that this is a three-part proof and so I'm not expecting any complete answers, but just some comments or tips in the right direction would help immensely.
For (a), I understand how it is true and I can sort of describe why ("multiples of factors greater than one will either be identical or will be separated by a distance greater than one, so if x is the factor then a(x)d–b(x)c will either equal 0 or have an absolute value greater than the absolute value of one"), but I'm not sure how to write this up or if it's totally accurate.
For (b) and (c) I am confused because from the examples I tried it seems like should be the one in between and and not . Also it seems like (c) should be a stipulation for (b) (and not just an addition to it), since there are pairs like and which are neighbor fractions but for which neither (b) nor (c) are true.
Any help is greatly appreciated!