I am self studying Apostol Dirichlet Series and Modular Functions in Number Theory and could not solve this question given in Chapter 7. Please help.
Question is – Let $ \Theta $ be an irrational number strictly between 0 and 1 . Then $ \Theta $ lies between two consecutive Farey fractions, say $ { a \over b } < \Theta < { c \over d } $.
Prove that either $ \Theta - { a \over b } < { 1 \over 2 b^2} $ or $ { c \over d } - \Theta < {1\over 2 d^2 } $.