Prove that any shape 1 unit area can be placed on a tiled surface

Question

Given a surface of equal square tiles where each tile side is 1 unit long. Prove that a single area A, of any shape, but just less than 1 unit square in area can be placed on the surface without touching a vertex of any tiled area? The Shape A may have holes.

Answer 1

Hint: Place $A$ randomly on the grid (consider the folding projection to just one square).

Answer 2

Project $A$ onto a single square by "Stacking" all of the squares in the plane. Then translating $A$ on this square corresponds to moving $A$ on a torus with surface area one. As the area of $A$ is less then one, there must be some point which it does not cover. Then choose that point to be the four corners of the square, and unravel the torus.