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[eestlinc]

Mo never asked for a proof, simply a solution.

Here's a proof-type question for you.

Say I start walking at the Cartesian coordinate (0,0) and I need to go to (1,1). I can walk at right angles, going to (0,1) then (1,1) for a distance of 2, but it would be shorter to go at 45 degrees straight to (1,1), for a distance of 1.414. But say I move infinitesimally north, to (0,delta), then infinitesimally east, to (epsilon,delta), then repeat this in a staircase pattern. As epsilon and delta go to zero, this path should represent the 45 degree path from (0,0) to (1,1). However, as long as epsilon and delta are finite, I will have to walk a distance of 2. Is it a contradiction that my right-angle approximation of the 45 degree path still results in a distance of 2? Is the 45 degree path really shorter?

Yes the 45 degree path is shorter. As long as your minute stairsteps have a definite length, each individual staircase is an exact replica of the full coordinate square between (0,0) and (1,1) and each stairstep can be more quickly traversed by taking the hypotenuse rather than both sides of the stairstep. The smaller the size of each stairstep, the more stairsteps are necessary to cover the full distance, and although each tiny stairstep is closer in length to the actual hypotenuse, say perahps a difference of .00001, you will have to travel 100000 of these steps to arrive at the point (1,1).

Or do you want a more formal proof?

X = horizontal distance traveled = 1
Y = vertical distance traveled = 1
X+Y = 2
k = number of steps (let's say for convenience that each step is the same size, although it doesn't matter whether they are or not)

distance of each step horizontal = D_h
distance of each step vertical = D_v

D_h = X/k
D_v = Y/k

D_{total per step} = X/k + Y/k

D_interval = D_{total per step} * k

D_interval = k(X/k + Y/k) = k(X/k) + k(Y/k) = x+y = 2

this is true regardless of the value of K and is thus independent of the value of K (the number of steps)

therefore:

lim_k-oo D_interval = lim_k-oo (X+Y) = lim_k-oo (2) = 2.