I do not agree with any of your statements  There is no such thing as an infinitesimal   is not an infinitesimal it is an indeterminate real value greater than  If this shocks you why does it not also shock you that pi is also indeterminate You can only approximate pi  Therefore      Tango There are infinitely many numbers between  and   check the archives and you will find a clear example You are incorrectly assuming that the limit of a series determines a number If you insist on defining a number by a series in fact this is circular you must be consistent what is the limit of any irrational number defined as a Cauchy sequence You do not know therefore you cannot use this definition Furthermore the apparatus for comparing numbers in decimal form must be consistent You start wih the most significant digit in the representation and continue the comparison until you find two digits that are not equal Having    contradicts this method You must also remember that it is impossible to represent all numbers in any radix system Most numbers can only be approximated And if this is so why would anyone foolishly insist on having