Exercise 3.  Verify Equations (5-3) and (5-4).  

Solution 3.

See text and/or instructor's solution manual.

Solution.   First prove  (5-3)   ,  if and only if  ,  where  n  is an integer.  

Let n be an integer, and set  .  

Then  .   Conversely, suppose  .  

Then  .   This implies  ,  this means that    for some integer  n.  

Since    is always positive and  ,  this means that    for some integer  n.  

And   implies and this forces  ,  so    .  

This establishes Property  (5-3) .

          Second,  prove  (5-4)   ,  if and only if   ,   for some integer  n.  

Property  (5-4)  comes from observing that     iff  ,   and appealing to Property (5-3). 

This solution is complements of the authors. 

(c) 2008 John H. Mathews, Russell W. Howell