Walking Distance Problem

Consider walking between two points A and B where the vertical and horizontal distance between them is 1km. Walking as shown in the following diagram would take 2km.

Now split up the trip into the 4 half km trips as shown below, the trip still takes 2 km to reach B from A.

Lets refine the trip even further into quarter km steps, the total distance travelled is still 2 km. Indeed, as we continue to split up the trip into smaller and smaller steps the total travel distance remains 2km.

However, as the step size reduced to zero, the travel path is along the diagonal and the travel distance is now suddenly only the square root of 2!

What's wrong with this argument, why does the travel distance suddenly drop from 2 to 1.414....?