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Also, a curve which gets arbitrarily close to every point in the unit square actually touches every point in the unit square. This is because (by definition) a

Also, a curve which gets arbitrarily close to every point in the unit square actually touches every point in the unit square. This is because (by definition) a curve is a continuous map from a compact space (the unit interval) to a Hausdorff space (R^2), and so its image is compact, and thus closed. A closed set contains every point that it is arbitrarily close to.

yarg 2y ago
If I travel one half of the distance from where I am to the finishing line an infinite number of times, I reach the finishing line but still never finish the race. With a Hilbert curve the entire plane becomes a limit.