Fermat’s Last Theorem

When in high school, spurred by Mr. Scheelbeek’s end-of-term inspirational lecture on Fermat’s Last Theorem, I tried proving the same for…about one and a half long years!
For documentation purposes, I’m attaching my proof. Feel free to outline the flaws in the comments section.

Let us assume FLT is true. i.e. $x^n + y^n =z^n$ . We know $x^n + y^n<(x+y)^n$ ( $n$ is assumed to be greater than one here). Hence, $z<x+y$ . Moreover, we know $z^n-x^n<(z+x)^n$ . Hence, $y<z+x$ . Similarly, $y+z<x$ .

So we have the three inequalities: $x+y<z, x+z<y,$ and $y+z<x$ .

$x,y,z$ satisfy the triangle inequalities! Hence, $x,y,z$ form a triangle.

Using the cosine rule, we get $z^2=x^2 +y^2 -2xy\cos C$ , where $C$ is the angle opposite side $z$ .

Raising both sides to the power $\frac{n}{2}$ , we get $z^n=(x^2 +y^2 -2xy\cos C)^{\frac{n}{2}}$ . Now if $n=2$ and $c=\frac{\pi}{2}$ , we get $z^2=x^2+y^2$ . This is the case of the right-angled triangle.