6.1 Definition Let be a given interval. By a partition of we mean a finite set of points where
Now suppose is a bounded real function defined on . Corresponding to each partition of we put
where the and are taken over all partition of . The left members of (1) and (2) are called the upper and lower Riemann integrals of over , respectively.
If the upper and lower integrals are equal, we say that is Riemann-integrable on , we write (that is, denotes the set of Riemann-integrable functions), and we denote the common value of (1) and (2) by
This is the Riemann integral of over . Since is bounded, there exist two numbers, and , such that
Hence, for every ,
so that the numbers and form a bounded set. This shows that the upper and lower integrals are defined for every bounded function .
 Rudin, W. (1976), Principles of Mathmatical Analysis (Third Edition), McGraw-Hill,Inc.