3.3 Relationship Between Monotone and Cauchy Sequences

Monotone and Cauchy sequences are key players in understanding convergence. Monotone sequences are either always increasing or decreasing, while Cauchy sequences have terms that get arbitrarily close to each other as we move along the sequence.

The relationship between these two types of sequences is fascinating. Bounded monotone sequences are always Cauchy sequences, which means they converge in complete metric spaces like the real numbers. This connection helps us prove important theorems and solve tricky problems involving limits.

Bounded Monotone Sequences as Cauchy Sequences

Definition and Properties of Monotone Sequences

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• A sequence $\{a_n\}$ is monotone if it is either monotonically increasing ($a_n \leq a_{n+1}$ for all $n$) or monotonically decreasing ($a_n \geq a_{n+1}$ for all $n$)
• Monotonically increasing sequences have terms that are non-decreasing ($1, 2, 3, 3, 4, \ldots$)
• Monotonically decreasing sequences have terms that are non-increasing ($5, 4, 4, 3, 2, \ldots$)
• A sequence $\{a_n\}$ is bounded if there exist real numbers $m$ and $M$ such that $m \leq a_n \leq M$ for all $n$
  • The sequence is bounded below by $m$ and bounded above by $M$
  • Example: The sequence $\{1/n\}$ is bounded below by $0$ and bounded above by $1$

Proving Bounded Monotone Sequences are Cauchy

• A sequence $\{a_n\}$ is a Cauchy sequence if for every $\varepsilon > 0$, there exists an $N \in \mathbb{N}$ such that $|a_n - a_m| < \varepsilon$ for all $n, m \geq N$
• To prove that every bounded monotone sequence is a Cauchy sequence, consider a bounded monotonically increasing sequence $\{a_n\}$ with bounds $m$ and $M$
  • For any $\varepsilon > 0$, choose $N$ such that $a_N > M - \varepsilon$. This is possible because $\{a_n\}$ is bounded above by $M$ and monotonically increasing
  • For any $n, m \geq N$, we have $a_N \leq a_n \leq a_m \leq M$, which implies $0 \leq a_m - a_n \leq M - a_N < \varepsilon$. Thus, $|a_n - a_m| < \varepsilon$ for all $n, m \geq N$, proving that $\{a_n\}$ is a Cauchy sequence
• A similar argument can be made for a bounded monotonically decreasing sequence
• Example: The sequence $\{1/n\}$ is monotonically decreasing and bounded, so it is a Cauchy sequence

Convergence of Monotone vs Cauchy Sequences

Convergence in Complete Metric Spaces

• In a complete metric space (such as $\mathbb{R}$), every Cauchy sequence converges to a limit within the space
• Since every bounded monotone sequence is a Cauchy sequence, it follows that every bounded monotone sequence converges in a complete metric space
• The Monotone Convergence Theorem states that a monotone sequence converges if and only if it is bounded
  • If a monotone sequence is unbounded, it diverges ($1, 2, 3, 4, \ldots$ diverges to $\infty$)
  • If a monotone sequence is bounded, it converges to a limit within the space ($1, 1/2, 1/3, 1/4, \ldots$ converges to $0$)

Limits of Convergent Monotone Sequences

• The limit of a convergent monotonically increasing sequence is the supremum (least upper bound) of the set of its terms
  • Example: The sequence $\{1 - 1/n\}$ is monotonically increasing and converges to $1$, which is the supremum of the set $\{1 - 1/n : n \in \mathbb{N}\}$
• The limit of a convergent monotonically decreasing sequence is the infimum (greatest lower bound) of the set of its terms
  • Example: The sequence $\{1/n\}$ is monotonically decreasing and converges to $0$, which is the infimum of the set $\{1/n : n \in \mathbb{N}\}$

Applications of Monotone and Cauchy Sequences

Determining Monotonicity and Boundedness

• Determine whether a given sequence is monotone (increasing or decreasing) by checking the inequality between consecutive terms
  • If $a_{n+1} \geq a_n$ for all $n$, the sequence is monotonically increasing
  • If $a_{n+1} \leq a_n$ for all $n$, the sequence is monotonically decreasing
• Verify if a monotone sequence is bounded by finding lower and upper bounds for the terms of the sequence
  • For a monotonically increasing sequence, the first term is a lower bound, and an upper bound can be found using the limit or other properties
  • For a monotonically decreasing sequence, the first term is an upper bound, and a lower bound can be found using the limit or other properties

Applying Convergence and Completeness Properties

• Use the Monotone Convergence Theorem to prove the convergence of a bounded monotone sequence
  • Example: Prove that the sequence $\{(1 + 1/n)^n\}$ converges by showing that it is monotonically increasing and bounded above by $e$
• Apply the definition of a Cauchy sequence to prove that a given sequence is Cauchy
  • Example: Prove that the sequence $\{1/n^2\}$ is Cauchy by finding an appropriate $N$ for any given $\varepsilon > 0$
• In a complete metric space, use the fact that every Cauchy sequence converges to solve problems involving the limit of a sequence
  • Example: In $\mathbb{R}$, find the limit of the sequence $\{1/n^2\}$ by using the fact that it is a Cauchy sequence and thus converges
• Prove the completeness of a metric space by showing that every Cauchy sequence in the space converges to a limit within the space
  • Example: Prove that $\mathbb{R}$ is complete by showing that any Cauchy sequence of real numbers converges to a real number
• Use the properties of monotone and Cauchy sequences to construct counterexamples or to prove statements about the convergence or divergence of sequences in various metric spaces
  • Example: Construct a monotonically increasing sequence in $(0, 1)$ that does not converge in $(0, 1)$ to show that $(0, 1)$ is not complete