Pythagorean Problem

PLEASE NOTE: The following work is presented as a mathematical puzzle. It is NOT a valid proof, but serves to illustrate the problems that can arise if one is not familiar with postulates and conditions of various theorems. Read it and try to find the problem, but PLEASE do not preach to the world that Pythagoras' Theorem is false.

A Disproof of Pythagoras' Theorem

The Theorem of Pythagoras

In a right triangle, the sum of the squares of the lengths of the two side sides is equal to the square of the hypotenuse.

a² + b² = c²

DISPROOF:
Start by defining a coordinate system with a along the x-axis and b along the y-axis. Let y = f(x) define the hypotenuse. Furthermore define a sequence of functions f_n(x) as 0 for x < a/n, b/n for a/n < x < 2a/n, ..., k b/n for k a/n < x < (k+1)a/n. ( A function with several steps).

For any given value of d > 0 , there is some value N such that for all n > N, b/n < d. Since |f(x) - f_n(x)| < b/n for all x, it follows that f_n(x) converges uniformly to f(x).

Clearly the length of the path defined by f₂(x) is a+b (or length a depending upon exactly how defines the path). Similarly, for any value of n the length of the path defined by f_n(x) is also a+b. Since the functions f_n(x) converge uniformly to f(x) the length of the path defined by f(x) is a+b.

Thus the Theorem of Pythagoras is incorrect. In reality, the length of the hypotenuse is equal to the sum of the lengths of the other two sides.

Some people who have tried to find the mistake argue about the functions being discontinuous at each step. However the sequence of functions can be slightly altered to produce continuous, infinitely differentiable functions which still satisfy the arguments. (Though they would triple the amount of work in the proof.)

PLEASE NOTE: The preceding work is presented as a mathematical puzzle. It is NOT a valid proof, but serves to illustrate the problems that can arise if one is not familiar with postulates and conditions of various theorems. Read it and try to find the problem, but PLEASE do not preach to the world that Pythagoras' Theorem is false.