Fraction work

WITH FULL SOLUTIONS PLEASE

Fractions a/b and c/d are called neighbor fractions if their difference (ad - bc)/(bd) has a numerator of positive or negative 1, that is, ad - bc = positive or negative one. Prove that

1. In this case neither fraction can be simplified (that is, neither has any common factors in numerator and denominator);

2. If a/b and c/d are neighbor fractions, then (a + b)/(c + d) is between them and is a neighbor fraction for both a/b and c/d; moreover,

3. no fraction e/f with positive integer "e" and "f" such that f is less than b + d is between a/b and c/d. [/img]