Let's call one direction for the rectangles edges "vertical" and the other "horizontal". Consider the vertical edge of the large rectangle. This edge is composed of several smaller edges of the tiling rectangles. Since this large edge is not a whole number, at least one of the vertical edges of the tiling rectangles must also be a non-whole number. Call this rectangle R. Since that rectangle's vertical edge is not a whole number, the other edge must be a whole number. That is, the horizontal edge. We may then draw a vertical line through the vertical edge of R that is not on the vertical edge of the large square to divide the large rectangle into two pieces, one of which does not contain R. Now, if the subrectangle that does not contain R is "trivial", in that the other edge of R lies on the other edge of the large rectangle, then the horizontal length of the large rectangle must be a whole number, which is a contradiction.
We now consider this smaller rectangle. The edges must not be whole numbers, so it is improper. The tiling on it inherited from the original rectangle must also be proper, since the changes in either of the edge lengths is by whole numbers. Finally, note that every rectangle in this tiling was present in the old tilling, and so the size of our tiling has decreased. Continuing this inductively, we must eventually arrive at an improper rectangle tiled by a single proper rectangle, which cannot happen.
An amazing illustration: