We have a square grid and we want to go from point $A$ to $B$, we can only go up or right. There's the first path depicted in red below, now let's half each step, so we get the blue path, halving again we get the green path, then the yellow and so on (Excuse the crappy drawing below, I did it with paint, each 'new direction' is supposed to be half of the previous one).
My question is, as the halving 'algorithm' tend to infinity, it appears closer and closer to the diagonal line which is $a\sqrt{2}\ $ but the path's total length can't be anything else but $2a$.
What is wrong here ? Why is the limit false ?
Note: suppose the grid is over $\mathbb R^2$.