The Real Numbers
This chapter concerns what can be thought of as the rules of the game: the
axioms of the real numbers. These axioms imply all the properties of the real
numbers and, in a sense, any set satisfying them is uniquely determined to
1. Basic Properties
Definition 3.1. A sequence is a function a : N R.
Instead of using the standard function notation of a(n) for sequences, it is
usually more convenient to write the argument of the function as a subscript, an .
In the end, all mathematics can be boiled down to logic and set theory. Because
of this, any careful presentation of fundamental mathematical ideas is inevitably
couched in the language of logic and sets. This chapter denes enough of
The Topology of R
1. Open and Closed Sets
Definition 5.1. A set G R is open if for every x G there is an > 0 such
that (x , x + ) G. A set F R is closed if F c is open.
The idea is that about every point of an open set, there is some room inside
Given a sequence an , in many contexts it is natural to ask about the sum of
all the numbers in the sequence. If only a nite number of the an are nonzero, this
is trivialand not very interesting. If an innite number of the terms arent zer