Chapter 2.7, Problem 16WE

To determine

To prove:

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

Explanation of Solution

Given information:

  $0<a<b$

Proof:

Here, the given proof statement is:

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

1. Given that:

  $0<a<b$

2. Here, use the first multiplication property for order:

  $ab>0$

3. Assume that:

  $a>0\Rightarrow \frac{1}{a}>0$

Here, use first multiplication property for order. Multiply by $\frac{1}{ab}$ on both sides of equation as:

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

4. Know that:

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

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

  $\frac{1}{b}<\frac{1}{a}$

5. Need to arrange the above inequality as:

  $\therefore \frac{1}{a}>\frac{1}{b}$

Hence proved