I am reading the book "Analysis I" from Terence Tao, and I have a doubt about an argument that I am using in my proof.
In first place, we already know this:
"Lemma 5.6.6 (a). If $x, y \geq 0$ are real numbers and $n \geq 1$ is a positive integer. Then $y^{n} = x$, if $y = x^{1/n}$."
The proposition I "proved" is the following:
Proposition. For each real number $x \geq 0$, $x^{1/1} = x$.
Proof. Let $x \geq 0$, a real number. Let $a := x^{1/1}$, and from Lemma 5.6.6 (a), we have $a^{1} = x$. We know $a^{1} = a$, and hence $a = x$. We can conclude $x^{1/1} = x$.
My question is basically if the argument in my proof when I define $a := x^{1/1}$, allows me to apply the Lemma 5.6.6 (a), and establish $a^{1} = x$. Is this correct?