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

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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.)

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