- #1

A.Magnus

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If $x \ne 0$, show that $\frac{1}{\frac{1}{x}} = x.$

The followings are what I was able to come out -- I just wanted to make sure that they are acceptable:

(a) By the multiplicative inverse property, there must exists $\frac{1}{x}$ such that $\frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = 1$.

(b) By the same property, there must exists $x$ such that $\frac{1}{x} \cdot x = 1$.

(c) By equating the (a) and (b) above, we have $\frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = \frac{1}{x} \cdot x$.

(d) By multiplying both sides with $x$, then we have $x \cdot \frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = x \cdot \frac{1}{x} \cdot x$, showing that $\frac{1}{\frac{1}{x}} = x$, as desired.

Thank you for your time and gracious helps. ~MA