Question about the proof of simple algebra rule $\frac{1}{\frac{1}{a}} = a$

Let $x=\frac{1}{a}$. Then:

\begin{eqnarray*} 1 = x \frac{1}{x} \Longrightarrow 1 = \frac{1}{a} \frac{1}{\frac{1}{a}} \Longrightarrow a = a \frac{1}{a} \frac{1}{\frac{1}{a}} \Longrightarrow \frac{1}{\frac{1}{a}} = a \end{eqnarray*}

Better? You're right, replacing $a\to\frac{1}{a}$ is a minor abuse of notation, because they're implicitly changing the variable without telling you. But you can fix that by just letting $x=\frac{1}{a}$ at the start.

$1/a$ is the multiplicative inverse $a^{-1}$ of

$a (\not =0)$, i .e. $a^{-1}a=1$.

Need to show:

$(a^{-1})^{-1} =a;$

Since

$(a^{-1})^{-1}(a^{-1})=1$;

$(a^{-1})^{-1}(a^{-1})a=1a=a$;

$(a^{-1})^{-1}(a^{-1}a)=a;$

$(a^{-1})^{-1}1=(a^{-1})^{-1}= a$.