If we consider a system of natural numbers without the existence of the number zero, one possible way to define an "approximation" of zero is to use a symbol, say * , to represent the absence of any natural number. This symbol can be treated as a new element, distinct from the set of natural numbers, and used to define arithmetic operations within the new system.

For instance, we can define addition and multiplication in this new system as follows:

- For any two natural numbers a and b, we define a + b as the (a + b)-th natural number. For example, 2 + 3 = 5, as usual.

- For any two natural numbers a and b, we define a * b as the * symbol, since there is no natural number corresponding to the product of a and b.

But how can we approximate zero using this system? One possible way to do it is to define a sequence of numbers (an) in this system such that this sequence approaches the symbol * as n grows. For example, we can define an=1/2^n, which would yield the sequence 1, 1/2, 1/4, 1/8, ..., all of which are natural numbers in this system. As n grows, the value of an approaches 0, but never actually reaches this value since there is no such natural number in the system. Thus, we can interpret the sequence {an} as an "approximation" of zero in this sense.