The theory of bounds is a fundamental starting point for almost all area of mathematics called analysis. And analysis is the study of functions of real and complex numbers.

If S is a non-empty subset of the set of real numbers R then there is an upper bound real number u for S such that the following inequality relation is always true: s ≤ u for all sÎS, this is saying that there is a real number s such that s is always less than or equal to u for every member s of the set S. Likewise, l is a lower bound for S if s³l for all sÎS. The existence of u and l can be used to define a least upper bound (LUB) and a greatest lower bound (GLB) for the subset S of R.