Bolzano–Weierstrass theorem

Bolzano–Weierstrass theorem In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space {displaystyle mathbb {R} ^{n}} . The theorem states that each bounded sequence in {displaystyle mathbb {R} ^{n}} has a convergent subsequence.[1] An equivalent formulation is that a subset of {displaystyle mathbb {R} ^{n}} is sequentially compact if and only if it is closed and bounded.[2] The theorem is sometimes called the sequential compactness theorem.[3] Contents 1 History and significance 2 Proof 3 Alternative proof 4 Sequential compactness in Euclidean spaces 5 Application to economics 6 See also 7 Notes 8 References 9 External links History and significance 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 proved again by Weierstrass. It has since become an essential theorem of analysis.

Proof First we prove the theorem for {displaystyle mathbb {R} ^{1}} (set of all real numbers), in which case the ordering on {displaystyle mathbb {R} ^{1}} can be put to good use. Indeed, we have the following result: Lemma: Every infinite sequence {displaystyle (x_{n})} in {displaystyle mathbb {R} ^{1}} has a monotone subsequence.

Proof: Let us call a positive integer-valued index {displaystyle n} of a sequence a "peak" of the sequence when {displaystyle x_{m}n} . Suppose first that the sequence has infinitely many peaks, which means there is a subsequence with the following indices {displaystyle n_{1}x_{n_{2}}>x_{n_{3}}>dots >x_{n_{j}}>dots } . So, the infinite sequence {displaystyle (x_{n})} in {displaystyle mathbb {R} ^{1}} has a monotone subsequence, which is {displaystyle (x_{n_{j}})} . But suppose now that there are only finitely many peaks, let {displaystyle N} be the final peak and let the first index of a new subsequence {displaystyle (x_{n_{j}})} be set to {displaystyle n_{1}=N+1} . Then {displaystyle n_{1}} is not a peak, since {displaystyle n_{1}} comes after the final peak, which implies the existence of {displaystyle n_{2}} with {displaystyle n_{1}

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