How about this for Problem 1. It's not as formal as it could be, but I'm not a pure mathematician by a long way.

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Consider a general square of dimension , centred on co-ordinates .

Now, by inspection (see the figure), the four possible locations of the centre of are:

where and are just multipliers taking the value or , depending on our choice.

If we now split that square similarly to form , the centre of that square will be at

with defined similarly. Extend this principle to theNth square, whose centre will be at

with and

The problem of whether the squares' centres will converge, as , is now reduced to the convergence (or not) of these two series.

It is not particularly troublesome that the sums contain the arbitrary sign-changing constants , because of a basic property of infinite series (assumed here) :

"The series converges if the series does."(ref: Absolute convergence .)

The series converges to unity. Hence, thex-andy-coordinates of converge to definite values, regardless of our choices of sign along the way. (QED!)