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Main article: Metric space In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance). Formally, A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function d \colon M \times M \rightarrow \mathbb{R} such that for any x, y, z \in M, the following holds: d(x,y) \ge 0 (non-negative), d(x,y) = 0\, iff x = y\, (identity of indiscernibles), d(x,y) = d(y,x)\, (symmetry) and d(x,z) \le d(x,y) + d(y,z) (triangle inequality) . Sequences and limits[edit] Main article: Sequence A sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers. One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distance between an and x approaches 0 as n → ∞, denoted |
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