Ge - ne - ral - lize - za - tion - s

It seems like I have proven that for $\forall \ x \in \mathbb R$ and $\forall \ y \in \mathbb N^*$ ; $x^y = 1$

Here's the prove:

Step 1: We have that $x^0 = 1$ .

Step 2: Assume that $x^k = 1; \forall \ 0 \le k \le y$ .

Step 3: We have that: $x^{y + 1} = \dfrac{x^y \cdot x^y}{x^{y - 1}} = 1$ This is true for $k = y + 1$ .

Step 4: Using inductive reasoning, the statement above is true for $\forall \ y \in \mathbb N^*$ .

I sent this prove to my teacher, and she said I was wrong at some point.

Where did I go wrong in this prove?

(This is actually a real story a while ago.)