Proof: Project the point P onto the x-y plane and call this projection P'. Project P' onto the x axis and call this projection P''. Now we have two right triangles to consider:
Let's consider the second of these first. Now, the leg of this triangle from the origin O to the point P'' has length |x_{1}| and the leg from P' to P'' has length |x_{2}|, so by the normal (two-dimensional) Pythagorean theorem, the hypotenuse of this triangle, which is the length of the segment from the orign to P', is
But the segment from P' to P has length |x_{3}|, so by applying the Pythagorean theorem once again, we get that
as desired.