Distance problem.

[LiveSsenkrad]

My friend has argued with another, that walking an alternative route would be faster than the usual route they follow. I am not able to provide values, but the rough sketch included shows my friend's theory. If you take the alternate road (the one with the 'triangles') and you make more triangles, just smaller, tending to infinity, you will eventually form a hypotenuse to the triangle, making that route obviously shorter. Is this flawed logic?

 

[Slapshot136]

it is flawed logic, as long as you still only move either horizontally or vertically at a time and not both, the route will stay the same length - and in real life, normally it takes longer to take a corner then it would be had you gone straight, therefore the smaller segments route would probably take longer