Cauchy’s Construction of R

π = 3.14159 . . . to mean that π is a real number which, accurate to 5 decimal places, equals the above string. To be precise, this means that |π − 3.14159| < 10−5. But this does not tell us what π is. If we want a more accurate approximation, we can calculate one; to 10 decimal places, we have π = 3.1415926536 . . . Continuing, we will develop a sequence of rational approximations to π. One such sequence is 3, 3.1, 3.14, 3.142, 3.1416, 3.14159, . . . But this is not the only sequence of rational numbers that approximate π closer and closer (far from it!). For example, here is another (important) one: