Exercise 67 Page 21

If we try to think about the greatest integer we can come up with, we immediately face a problem. We just cannot! No matter what number we choose, we can always think of an integer that could come next. Then, the same thing happens with the new one. Let's look at an example. 100, 000 < 100, 000 + 1 < 100, 000 +2 < ... The numbers never end. There cannot be a greatest one. Something similar happens when looking for the least fraction. Take a look at the following succession. 1/2 > 1/3> 1/4 > 1/5 ... Each time the value in the denominator gets bigger, the fraction becomes less than the previous one. The least fraction would be the one with the greatest integer as the denominator. But, as we just said, integer numbers never reach an end. Therefore, there is no greatest integer to be in the denominator. This means that there cannot be a least fraction either. We will be always able to find one less than our original choice.