Say you want to move from point A to point B. You can go directly between the points, or you can move in the x direction, then the y direction, like so:

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`A----------x`

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B

Lousy drawing, sorry. Obviously the direct path is shortest here. You could also move halfway in the x direction, then halfway in y, then the rest of the way in x and rest of the way in y, so that the path looks like a step. Theoretically that is the same distance as going the whole way in x, then the whole way in y. If you keep reducing the interval between each direction change (say 1/3, then 1/4, then 1/10, etc), it should still remain the same distance as the first path, right? What happens when the interval goes to zero and the path is effectively the same as the direct path? Shouldn't it be shorter?