Chapter 2.7, Problem 18WE

To determine

To prove: For every $a>0$ , $a>1$ .

Explanation of Solution

Given information: $a>\frac{1}{a}$ .

Proof: Since $a>0$ , so it is a positive number.

Let $a=1$ ,

Multiplying both sides by $\frac{1}{a}$ , we get,

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

Here, it contradicts the fact that $a>\frac{1}{a}$ ,

Hence, $a\ne 1$

Now, assume that $a<1$ ,

Since $a$ is positive, $\frac{1}{a}$ is also be positive.

Multiply both sides of $a<1$ by $\frac{1}{a}$ , we have,

  $\begin{array}{l}1<\frac{1}{a}\\ a<1<\frac{1}{a}\end{array}$

Here, it contradicts the fact that $a>\frac{1}{a}$ .

Thus, both assumptions are wrong.

Therefore, $a>1$