i reply number two in order to correct my two previous thread (there is indeed a problem whith that if anyone has got a solution?)
last one was stupid i could'nt cancel it
the one before i was speaking about what we call in france and probably everywhere bolzano-weistrat property (banach is the nature of the 'closed' 'space' thath admit that property)nce you have extracted a one-at-least convergent serie you just have to proove that your serie Sn converge to the same point

but lets try to do number two
if S and T would not be steady by multiplication (excuse my poor memory)
there would be As and Bs and Cs belonging to S
and At and Bt and Ct belonging to T

with As*Bs=Ct and At*Bt=Cs
because all this number belongs to U wich is steady by multiplication and because S and T are forming a partition of U
so in U were allowed to write As*Bs*Cs=At*Bt*Ct
so if the property that the multiplication of tree elements of the two subset belongs to the subset S and T they therefor have a comon element and cannot form a partition!!!