Cauchy sequences and completeness are crucial concepts in mathematical analysis. They help us understand how sequences behave in different spaces and provide a way to characterize convergence without relying on limits.
These ideas build on our understanding of monotone sequences, extending our toolkit for analyzing sequence behavior. By studying Cauchy sequences and completeness, we gain deeper insights into the properties of real numbers and other metric spaces.
Cauchy sequences in analysis
Definition and properties
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A sequence {xn} in a metric space (X, d) is a Cauchy sequence if for every ε > 0, there exists an N ∈ ℕ such that for all m, n ≥ N, d(xm, xn) < ε
The definition of a Cauchy sequence describes a property of the terms of the sequence getting arbitrarily close to each other as the sequence progresses
Cauchy sequences are named after the French mathematician Augustin-Louis Cauchy, who introduced the concept in the early 19th century
Significance in mathematical analysis
Cauchy sequences are fundamental in mathematical analysis because they provide a way to characterize completeness, a crucial property of the real numbers and other complete metric spaces
In a complete metric space, every Cauchy sequence converges to a limit within the space
This property is essential for proving the existence of limits and for constructing new mathematical objects
The definition of a Cauchy sequence does not require the existence of a limit; it only describes a property of the terms of the sequence getting arbitrarily close to each other as the sequence progresses
Convergent sequences vs Cauchy sequences
Proving convergent sequences are Cauchy
To prove that every convergent sequence is a Cauchy sequence, start by assuming that {xn} is a sequence in a metric space (X, d) that converges to a limit x ∈ X
By the definition of convergence, for every ε > 0, there exists an N1 ∈ ℕ such that for all n ≥ N1, d(xn, x) < ε/2
For m, n ≥ N1, apply the triangle inequality to show that d(xm, xn) ≤ d(xm, x) + d(x, xn) < ε/2 + ε/2 = ε
This proves that {xn} satisfies the definition of a Cauchy sequence, as for every ε > 0, there exists an N1 ∈ ℕ such that for all m, n ≥ N1, d(xm, xn) < ε
Converse not always true
The converse of the statement "every convergent sequence is a Cauchy sequence" is not always true
A Cauchy sequence may not always converge in a given metric space
The space in which every Cauchy sequence converges is called a complete metric space
Examples of complete metric spaces include the real numbers (ℝ) and the complex numbers (ℂ)
An example of an incomplete metric space is the rational numbers (ℚ), where the sequence {1, 1.4, 1.41, 1.414, ...} is Cauchy but does not converge to a rational number
Completeness of the real numbers
Demonstrating completeness using Cauchy sequences
To demonstrate the completeness of the real numbers, show that every Cauchy sequence of real numbers converges to a real number
Start by considering a Cauchy sequence {xn} in ℝ
By the definition of a Cauchy sequence, for every ε > 0, there exists an N ∈ ℕ such that for all m, n ≥ N, |xm - xn| < ε
Use the Cauchy sequence {xn} to construct two sequences {an} and {bn}, where an = inf{xk : k ≥ n} and bn = sup{xk : k ≥ n}
Prove that these sequences converge to the same limit L
Monotone Convergence Theorem and Cauchy sequences
Show that {an} is monotonically increasing and bounded above by xN + ε, and {bn} is monotonically decreasing and bounded below by xN - ε
Apply the Monotone Convergence Theorem to conclude that {an} and {bn} converge to limits, say L1 and L2, respectively
Prove that L1 = L2 by assuming the contrary and deriving a contradiction using the properties of Cauchy sequences
Conclude that the Cauchy sequence {xn} converges to the real number L, thus demonstrating the completeness of ℝ
Identifying Cauchy sequences
Analyzing the behavior of terms
To determine whether a given sequence {xn} in a metric space (X, d) is a Cauchy sequence, analyze the behavior of the terms as n and m approach infinity
Find a general expression for d(xm, xn) and then investigate its behavior as m and n grow arbitrarily large
If d(xm, xn) approaches 0 as m and n approach infinity, the sequence is a Cauchy sequence (e.g., {1/n} in ℝ)
If d(xm, xn) does not approach 0 or grows unbounded as m and n approach infinity, the sequence is not a Cauchy sequence (e.g., {n} in ℝ)
Using the definition directly
Another approach is to use the definition of a Cauchy sequence directly
Assume the sequence is a Cauchy sequence and try to find an N ∈ ℕ for an arbitrary ε > 0 such that for all m, n ≥ N, d(xm, xn) < ε
If successful, the sequence is a Cauchy sequence (e.g., {1 + 1/n} in ℝ)
If unable to find such an N for some ε > 0, the sequence is not a Cauchy sequence (e.g., {1 - 1/n} in ℚ)
Examples of Cauchy and non-Cauchy sequences
Examples of sequences that are Cauchy sequences include convergent sequences, such as {1/n} in ℝ, and sequences in complete metric spaces, such as {1 + 1/n} in ℝ
Examples of sequences that are not Cauchy sequences include divergent sequences, such as {n} in ℝ, and sequences in incomplete metric spaces, such as {1 - 1/n} in ℚ