Real numbers are defined as the limits of convergent series of rational numbers.

A = lim(n->oo){sum(i=1,n,a_i/10^i)}, with a_i = 0,1,...,or 9.

Defines a number in [0,1].

To expand,

For all epsilon > 0, there exists N(epsilon) such that n > N(epsilon) implies |sum(i=1,n,a_i/10^i) - A| < epsilon.

Two questions.

1) What is the meaning of |sum(i=1,n,a_i/10^i) - A| ? The difference between a rational number "sum(i=1,n,a_i/10^i)" and a number that is being defined "A" ?

2) What number system does epsilon come from ? The rational numbers or the number system we are defining ? If the rational numbers, then why not use 1/m for some m > 0 ?