Let a and b be real numbers. Prove or disprove each of the following. In your proofs you may assume that the product of two positive real numbers is positive, the product of a negative and a positive number is negative, the product of two negative numbers is positive.
If a < b and c > 0, then ac < bc.
If a < b and c < 0, then ac > bc.
If 0 < c < 1, then ac < a.
For any positive real number c, |a-b| < c iff b is in (a-c, a+c).
For any positive real number c, |a-b| < c iff a is in (b-c, b+c).
For any positive real number c, |a-b| < c iff |a-c| < b.