Ĭonvergence actually means that the corresponding sequence gets as close as it is desired without actually reaching its limit. if there is an integer such that whenever. Īccordingly, a real number sequence is convergent if the absolute amount is getting arbitrarily close to some (potentially unknown) number, i.e. Those points are sketched smaller than the ones outside of the open ball. an infinite number) smaller than lie within the open ball. For instance, for we have the following situation, that all points (i.e. Note that represents an open ball centered at the convergence point or limit x. We can illustrate that on the real line using balls (i.e. A sequence that fulfills this requirement is called convergent. We mean that for every real number there is an integer, such that Now, let us try to formalize our heuristic thoughts about a sequence approaching a number arbitrarily close by employing mathematical terms. It is only important that the sequence can get arbitrarily close to its limit. Note that it is not necessary for a convergent sequence to actually reach its limit. Does the sequence approach a (real) vector/number?.We are interested in what the long-term behavior of the sequence is: Other sequences may not give us that impression. number) that are “ getting close” to some other vector (e.g. We might think that some of the above examples contain patterns of vectors (e.g. Plot of 2-tuple sequence for the first 1000 points that seems to head towards a specific point in. Plot of the sequenceĮ) Consider the 2-tuple sequence in. Here, each natural is mapped on itself.ĭ) Consider, that can also be written as. The range of the function only comprises two real figures. Plot of forī) Let us now consider the sequence that can be denoted by. Sequences are, basically, countably many ( – or higher-dimensional) vectors arranged in an ordered set that may or may not exhibit certain patterns.Ī) The sequence can be written as and is nothing but a function defined by. Latter concept is very closely related to continuity at a point. We need to distinguish this from functions that map sequences to corresponding function values.
Note that a sequence can be considered as a function with domain. Property holds for almost all terms of if there is some such that is true for infinitely many of the terms with. Let be a -tuple sequence in equipped with property. Note that latter definition is simply a generalization since number sequences are, of course, -tuple sequences with. Sequences in, are called real tuple sequences. Sequences in are called real number sequences. The elements of are called terms of the sequence. If the metric is not specified, we assume that the standard Euclidean metric is assigned.Ī sequence in is a function from to by assigning a value to each natural number. We thereby restrict ourselves to the basics of limits. In this section, we apply our knowledge about metrics, open and closed sets to limits.
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