Say we're walking on the unit square and wish to make our way from (0, 0) to (1, 1). Obviously, we can go one unit up and one unit right, or, for the same distance, one unit right and one unit up. Similarly, we can go half a unit up and then half a unit right and then repeat these steps, or the mirrored alternative. And so on...
Each of the above options, as long as we allow ourselves to travel only finite distances and only in one axis direction at a time, will result in the same final distance, namely 2. Yet in the limit, these steps will have to collapse to $\sqrt{2}$, as we will by this point be traveling just on the diagonal.
First, are my above intuitions correct? And second, if so, how can we demonstrate the above to be the case?
