What two perimeters? There’s only one. And it hasn’t been shown to bound the triangle’s perimeter from below. It is plainly greater than the triangle’s perimete

What two perimeters? There’s only one. And it hasn’t been shown to bound the triangle’s perimeter from below. It is plainly greater than the triangle’s perimeter at every step, so it just establishes an upper bound. To derive the triangle’s perimeter, you also need another thing that’s always less than the triangle’s perimeter, then show that the lower and upper bounds converge on the same value.