Bolzano–Weierstrass theorem

In real analysis, the Bolzano-Weierstrass theorem is an important theorem characterizing sequentially compact sets.

Statement

In actual analysis, the Bolzano-Weierstrass theorem is a fundamental result concerning convergence in a finite-dimensional Euclidean space Rⁿ. The theorem states that every bounded sequence in Rⁿ has a convergent subsequence. An equivalent formulation is that a subset of Rⁿ is sequentially compact if and only if it is closed and bounded.

Demo

First, applying the method of mathematical induction, we will prove the theorem when n = 1, in which case the ordering of R can be put to good use. In fact we have the following result.

Motto: Each succession { x_n } in R has a monotonic subsequence.

Demonstration: Let's call a positive integer n a "pic of the sequence", if m/2005 n implies x_n ▪ x_mI mean, yeah. x_n is greater than all the following terms of succession. Let us first suppose that succession has infinite peaks, n₁. n₂. n₃... n_jthen the corresponding subsidization {xnj!{displaystyle {x_{n_{j}}}}}to peaks is monotonously decreasing, so the motto is tested. So let's suppose now that there's only a finite number of beaks, be it N the last beak, and suppose a new succession {xnj!{displaystyle {x_{n_{j}}}}}Where n₁ = N + 1. Later. n₁ is not a beak, since n₁ ▪ Nwhich implies the existence of a n₂ ▪ n₁ with xn2≥ ≥ xn1.{displaystyle x_{n_{2}}geq x_{n_{1}}}}. !Once again, n₂ ▪ N It's not a beak, so there's n₃ ▪ n₂ with xn3≥ ≥ xn2.{displaystyle x_{n_{3}}geq x_{n_{2}}}}. !Repeating this process leads to an ever-increasing infinite subsidization xn1≤ ≤ xn2≤ ≤ xn3≤ ≤ ...... {displaystyle x_{n_{1}}{ldots }If you wish.

Now suppose we have a bounded sequence in R, by the Lemma there exists a monotonic subsequence, necessarily bounded. But it follows from the monotonic convergence theorem that this subsequence must converge, and the proof is complete. Finally, the general case can be easily reduced to the case of n = 1 as follows: given a bounded sequence in Rⁿ, the sequence of the first few coordinates is a limited real sequence, therefore it has a convergent subsequence. You can then extract a subsequence in which the second coordinates converge, and so on, until finally we have gone from the original sequence to a subsequence n times—which is still a subsequence of the original sequence—in which each coordinate converges. sequence; therefore, the subsequence itself is convergent.

Sequential compactness in Euclidean spaces

Suppose A is a subset of Rⁿ with the property that every sequence in A has a subsequence convergent to an element of A. Then A must be limited, otherwise there exists a sequence in the x_m in A with || x_m || ≥ m for all m, and then each subsequence is unbounded and therefore non-convergent. On the other hand A must be closed, since from a point of non-interior x in the complement of A an sequence can be constructed >A with values of convergence to x. Thus, the subsets A, of Rⁿ, so that each sequence in A has a subsequence convergent to an element of A—that is, the subsets that are sequentially compact in the subspace topology—are precisely the closed and bounded sets. This form of the theorem makes the analogy with the Heine-Borel Theorem especially clear, which states that a subset of Rⁿ is compact if and only if it is closed and bounded. In fact, the general topology tells us that a space is compact metrizable if and only if it is sequentially compact, so that the Bolzano-Weierstrass and the Heine-Borel theorem are essentially the same.

History

The Bolzano-Weierstrass theorem is named after mathematicians Bernard Bolzano and Karl Weierstrass. It was actually first proved by Bolzano in 1817 as a lemma in the proof of the intermediate value theorem. Some fifty years later, the result was identified as significant in its own right, and demonstrated once more by Weierstrass. Since then it has become a fundamental theorem of analysis.