![]() ![]() A convergent subsequence exists for every sequence in a closed and bounded set S in R n.A convergent subsequence exists for any bounded sequence in R n.A convergent subsequence exists for every bounded sequence of real numbers.Bolzano Weierstrass Theorem for Sequences ![]() It is, in fact, equivalent to the real-number completeness axiom. Many analytical results are based on the Bolzano Weierstrass theorem. The Bolzano Weierstrass theorem is a theorem that states that a convergent subsequence, or subsequential limit, exists for every bounded sequence of real numbers. (Throughout this section, we’ll assume that R n has a norm we’ve already demonstrated that the norm we employ in R n has no bearing on convergence - that is, on which sequences converge.) Bolzano Weierstrass Theorem The Bolzano-Weierstrass Theorem is about this. We now look at an easy-to-prove condition that ensures that a sequence in R or R n has a convergent subsequence: any limited sequence in Rn has a Cauchy subsequence, which converges in R n. When does a Sequence have a Convergent Subsequence or vice versa?Ĭonsider the alternating real sequences. In this article, we will discuss all the Bolzano Weierstrass theorems. In reality, the only ones that do converge are those that are “very good.” Even “excellent” sequences, however, may fail to converge.įor example, Cauchy sequences are excellent they’re not much different from convergent sequences in that they converge in “good” spaces (i.e., complete spaces) and fail to converge only when the point that “should be” its limit is not in the space in other words, it fails to converge because the space is not good (it’s incomplete), not because the sequence is bad. ![]()
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