Bachelier said:One of the intervals, [a, (a+b)/2] or [(a+b)/2, b] contains an infinite set of members of S.
The Bolzano-Weierstrass proof by construction is a mathematical proof that shows every bounded sequence has a convergent subsequence. It is named after mathematicians Bernard Bolzano and Karl Weierstrass.
The proof works by dividing the interval containing the sequence into smaller and smaller subintervals, and then selecting a value from each subinterval to create a subsequence. This subsequence will converge to a limit point, which is also the limit of the original sequence.
This proof is important because it is a fundamental result in real analysis and is used in many other areas of mathematics, such as calculus, differential equations, and topology. It also provides a method for constructing convergent sequences, which is useful in many other mathematical proofs.
No, the proof only applies to bounded sequences. If a sequence is unbounded, it may not have a convergent subsequence. In this case, the Bolzano-Weierstrass proof cannot be used.
The proof has applications in physics, engineering, and economics, where it is used to analyze and model various systems. It is also used in computer science and data analysis to optimize algorithms and find solutions to complex problems.