Sequence b) instead is alternating between and and, hence, does not converge. Let us re-consider Example 3.1, where the sequence a) apparently converges towards. Let for then the assertion follows immediately. We can prove this intuitive statement by setting. Notice, that a ‘detour’ via another convergence point (triangle property) would turn out to be the direct path with respect to the metric as. While a sequence in a metric space does not need to converge, if its limit is unique. The convergence of the sequence to 0 takes place in the standard Euclidean metric space. The definition of convergence implies that if and only if. For instance, the sequence Example 3.1 a) converges in to 0, however, fails to converge in the set of all positive real numbers (excluding zero). It might be that a sequence is heading to a number that is not in the range of the sequence (i.e. Please note that it also important in what space the process is considered. If there is no such, the sequence is said to diverge. The sequence can also be considered as a function defined by with “Arbitrarily close to the limit ” can also be reflected by corresponding open balls, where the radius needs to be adjusted accordingly. This limit process conveys the intuitive idea that can be made arbitrarily close to provided that is sufficiently large. Hence, it might be that the limit of the sequence is not defined at but it has to be defined in a neighborhood of. 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.
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