Limit of orthogonal lines to straight line help?

[Jonnyb42]

Yesterday I thought of a math problem, and it seems very simple, as I assume the solution is, and I want to know the answer more than I want to figure it out myself.

Ok imagine point A and point B. The shortest path from A to B is a straight line. Let's now go from A to B in two orthogonal lines. I like to think of a right triangle as the two different paths, (hypotenuse is the shortest path and the legs form the longer path.) Now, if you take yet another path but in 4 lines, and then in 8 lines, 16 lines, and on and on, you will eventually be matching the shortest path, or the hypotenuse, however the problem is, the total distance traveled in each successive path is still the same, how does it all of a sudden get to the shortest? In other words the limit as the number of component paths you take, n (n being even and and in the pattern, 2,4,8,16... required geometrically) goes to infinity of the total distance travelled, the path seems to go to the hypotenuse, yet mathematically remains the sum of the two legs, making it seem like (if each leg is 1 unit) 2 = sqrt(2) which of course is wrong.

http://reshall.site11.com/dia1.bmp

 

[Hurkyl]

The stair-step paradox is a classic! Basically, it amounts to a proof that in no mathematical structure can the following two statements both be true:

1. The sequence of stair-steps converges to the diagonal segment
2. Length is a continuous function of one-dimensional shapes

In a structure where (1) is true, this becomes a classic error of interchanging a limit with another operation. You computed two numbers:

1. You took the limit of shapes, then found the length
2. You found the lengths, then took the limit of numbers

There is no reason to think the two calculations should give the same result, aside from the fact many useful operations are allowed to be interchanged with limits. Alas, this is not one of them.