the real number line. The order properties of the real numbers described above are closely related to these topological properties. As a topological space, the real numbers has a standard topology, which is the order topology induced by order . Alternatively, by defining the metric or distance function using the absolute value function as the real numbers become the prototypical example of a metric space. The topology induced by metric turns out to be identical to the standard topology induced by order . Theorems like the intermediate value theorem that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods. Sequences A sequence is a function whose domain is a countable, totally ordered set. The domain is usually taken to be the natural numbers, although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices. Of interest in real analysis, a real-valued sequence, here indexed by the natural numbers, is a map . Each is referred to as a term (or, less commonly, an element) of the sequence. A sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses: A sequence that tends to a limit (i.e., exists) is said to be convergent; otherwise it is divergent. (See the section on limits and convergence for details.) A real-valued sequence is bounded if there exists such that for all . A real-valued sequence is monotonically increasing or decreasing if or holds, respectively. If either holds, the sequence is said to be monotonic. The monotonicity is strict if the chained inequalities still hold with or replaced by < or >. Given a sequence , another sequence is a subsequence of if for all positive integers and is a strictly increasing sequence of natural numbers. Limits and convergence Roughly speaking, a limit is the value that a function or a sequence "approaches" as the input or index approaches some value. (This value can include the symbols when addressing the behavior of a function or sequence as the variable increases or decreases without bound.) The idea of a limit is fundamental to