for probleme 1 i have got an idea that i don't know if i will be able to translate it in english

topoligicaly speeking a square is a 'closed' part of the space (un 'espace fermé' or 'fermé' in french wich as a part of R*R admit the 'banach property "every infinite serie of point of it admit an convergent 'extraded' serie (a convergent infinite serie constructed by forgoting some points of the serie from wich it is extracted)
it will be relatively easy to proove that every extracted series that you can build will converge to points not more distant that a predefinite epsilon of your choice
that's just an idea i'm essentialy trying my english and if i would like to be a litle bit more rigourous i would need to take a sheet of paper and a boock