4. Prove that for each a E R and each n E N there exists a rational rn such that la - rnl < lin.

r:m. [ApPROXIMATION PROPERTY FOR INFIMA] This exercise is used in many sections, including 2.2 and 5.1.

(a) By modifying the proof of Theorem 1.20, prove that if a set E c R has a finite infimum and f > 0 is any positive number, then there is a point a E E such that inf E + f > a ~ inf E.

(b) Give a second proof of the Approximation Property for Infima by using Theorem 1.28.