What's wrong with this induction based proof?

Question

Claim:
$\forall x \in \mathbb{R^+} ,$ $ x^n=1 $ $where$ $ n\in \mathbb{N}$

Proof by induction on n:

Basis step:
$\forall x \in \mathbb{R^+} ,$ $ x^0=1 $

Induction Step:
Let this holds for all n$\lt n_0$
then $x^{n_0}$=$\frac{x^{n_0-1}}{x^{n_0-2}} \times x^{n_0-1} $=$\frac{1}{1} \times1 $ $\square $

Answer 1

If you apply it to the case that $n_0=1$ you will have a problem applying the induction hypothesis to $n_0-2=-1$, what with it not being a natural number.

Answer 2

Check how to use your indcution to prove $x=1, \forall x\in R+$

If you could prove $x^{-1} = 1, \forall x\in R^+$, then you could get your conclusion, but...