[londons_explore]

Why it works:

Imagine what would be required to put a dot in an area that ought to be blank... The previous point would have had to have been outside the triangle, or in another blank area at the previous step.

Hence, it is impossible to draw in the blank areas.

[guimplen]

I think your argument only shows that k-th point lies inside the k-th approximation of the Sierpinsky triangle, but says nothing abiut the fact that the final picture loooks like the Sierpisnsky triangle.