Chapter 2.7, Problem 15WE

To determine

To prove:

If $a>0,b>0$ and $\frac{1}{a}>\frac{1}{b}$ then $a<b$

Explanation of Solution

Given information:

  $\begin{array}{l}a>0\\ b>0\\ \frac{1}{a}>\frac{1}{b}\end{array}$

Proof:

Here, the given proof statement is:

If $a>0,b>0$ and $\frac{1}{a}>\frac{1}{b}$ then $a<b$

1. $a>0,b>0$ and $\frac{1}{a}>\frac{1}{b}$ : This is the given in the problem.
2. Here, use the first multiplication property for order. On the both sides of the given inequality equation need to multiply by $ab$

  $ab\cdot \frac{1}{a}>ab\cdot \frac{1}{b}$

Here $ab$ is positive as $a>0,b>0$

3. Use the property of reciprocals as:

  $\begin{array}{l}a\cdot \frac{1}{a}=1\\ b\cdot \frac{1}{b}=1\end{array}$

Get that:

  $b>a$

4. Use the property of multiplicative inverses to get the reciprocals on both the sides of equation:

  $\frac{1}{b}<\frac{1}{a}$

5. Multiply both the sides of equation by $ab$ as:

  $\frac{1}{b}\cdot ab<\frac{1}{a}\cdot ab$

6. The property of reciprocals is used to get the final result as:

  $\therefore a<b$

Hence proved.