sequence, or doubly infinite sequence. A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ), is bi-infinite. This sequence could be denoted . Increasing and decreasing A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For example, the sequence is monotonically increasing if and only if an+1 an for all n ∈ N. If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonically decreasing if each consecutive term is less than or equal to the previous one, and is strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function. The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively. Bounded If the sequence of real numbers (an) is such that all the terms are less than some real number M, then the sequence is said to be bounded from above. In other words, this means that there exists M such that for all n, an ≤ M. Any such M is called an upper bound. Likewise, if, for some real m, an ≥ m for all n greater than some N, then the sequence is bounded from below and any such m is called a lower bound. If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded. Subsequences A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of the positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved. Formally, a subsequence of the sequence is any sequence of the form , where is a strictly increasing sequence of positive integers. Other types of sequences Some