KAJIAN TEOREMA BOLZANO-WEIERSTRASS UNTUK MENGKONSTRUKSI BARISAN YANG KONVERGEN DI R^n DAN APLIKASINYA DALAM PEMBUKTIAN TEOREMA EKSISTENSI MAX-MIN

Abstract

A sequence is a function from the set of natural numbers to the set of real numbers . In sequences there is the concept of sequence convergence. Testing the convergence of a sequence can be done using the Bolzano-Weierstrass Theorem. This theorem states that every finite sequence has a convergent sequence. The relationship between convergent sequences and finite sequences is also important to study further. Apart from being used to prove the convergence of sequences, the Bolzano-Weirstrass Theorem can also be applied to prove the Max-Min Existence Theorem. This research was conducted to examine the relationship between convergent sequences and finite sequences, the relationship between convergence and continuous functions and the relationship between continuous functions and max-min values ​​with the aim of constructing a convergent sequence in R^n and its application in proving the Max-Min Existence Theorem. This research is a literature study. This research was conducted through a literature review of books and other literature. From the literature review, the materials are then discussed in depth. The results of the literature study show that a convergent sequence is a finite sequence, but a finite sequence is not necessarily convergent. In determining the convergence of a sequence using the Bolzano-Weierstrass Theorem, it is necessary to first show the limitations of the sequence. Furthermore, to prove the Max-Min Existence Theorem it is necessary to require that the sequence is finite and then this theorem can be proven using the Bolzano-Weierstrass Theorem and Apit Theorem.

Keywords: Monotonous Sequence, Finite Sequence, Continuity, Bolzano-Weierstrass Theorem, Max-Min Existence Theorem.