Definition P3.1. Let and be metric spaces; suppose , maps into , and is a limit point of . We write as , or
if there exists with the following property: for every there exists such that
for all points for which . Here and refer to distances in and , respectively. If and/or are replaced by (euclidean space), then , are replaced by absolute differences, or norms of differences.
”Given a function that maps the sequence that converges at in to a point in called , we define the following notation: .”
Theorem P3.2. Let ,,,, and be as in the previous definition. Then
for every sequence in such that ,.
Corollary P3.3. If has a limit at , the limit is unique.
Definition P3.4. Suppose we have two complex functions, and , both defined on . By we mean the function which assigns to each point of the number . Similarly we define the difference, the product, and the quotient of the two functions, with the understanding that the quotient is defined only at those points of at which .