🏃🏽‍♀️‍➡️Intro to Mathematical Analysis Unit 3 – Monotone and Cauchy Sequences

Key Concepts and Definitions

• Monotone sequences are sequences that either increase or decrease consistently
  • Increasing monotone sequence: $a_n \leq a_{n+1}$ for all $n \in \mathbb{N}$
  • Decreasing monotone sequence: $a_n \geq a_{n+1}$ for all $n \in \mathbb{N}$
• Strictly monotone sequences have strict inequalities ($<$ or $>$) instead of $\leq$ or $\geq$
• Bounded sequences have an upper bound and/or a lower bound
  • Upper bound: $\exists M \in \mathbb{R}$ such that $a_n \leq M$ for all $n \in \mathbb{N}$
  • Lower bound: $\exists m \in \mathbb{R}$ such that $a_n \geq m$ for all $n \in \mathbb{N}$
• Convergent sequences approach a limit as $n$ approaches infinity
  • Limit of a sequence: $\lim_{n \to \infty} a_n = L$ if $\forall \varepsilon > 0, \exists N \in \mathbb{N}$ such that $|a_n - L| < \varepsilon$ for all $n \geq N$
• Cauchy sequences have terms that become arbitrarily close to each other as $n$ increases
  • Cauchy sequence: $\forall \varepsilon > 0, \exists N \in \mathbb{N}$ such that $|a_n - a_m| < \varepsilon$ for all $n, m \geq N$

Properties of Monotone Sequences

• Monotone sequences are either non-increasing or non-decreasing
• Every monotone sequence is either convergent or divergent
  • Convergent if bounded (monotone convergence theorem)
  • Divergent if unbounded ($\lim_{n \to \infty} a_n = \infty$ for increasing, $\lim_{n \to \infty} a_n = -\infty$ for decreasing)
• The limit of a convergent monotone sequence is equal to its supremum (increasing) or infimum (decreasing)
  • Supremum: least upper bound, $\sup\{a_n\} = \min\{x \in \mathbb{R} : a_n \leq x \text{ for all } n \in \mathbb{N}\}$
  • Infimum: greatest lower bound, $\inf\{a_n\} = \max\{x \in \mathbb{R} : a_n \geq x \text{ for all } n \in \mathbb{N}\}$
• Arithmetic operations preserve monotonicity
  • Sum, difference, product, and quotient of monotone sequences are monotone (assuming divisor sequence is never zero)
• Subsequences of monotone sequences are also monotone

Convergence of Monotone Sequences

• Monotone convergence theorem states that a monotone sequence converges if and only if it is bounded
  • Increasing and bounded above $\implies$ convergent
  • Decreasing and bounded below $\implies$ convergent
• The limit of a convergent monotone sequence can be found using the supremum or infimum
  • $\lim_{n \to \infty} a_n = \sup\{a_n\}$ for increasing sequences
  • $\lim_{n \to \infty} a_n = \inf\{a_n\}$ for decreasing sequences
• Convergence of monotone sequences can be proven using the definition of convergence and the monotone convergence theorem
• Rate of convergence for monotone sequences depends on the specific sequence
  • Geometric sequences ($a_n = ar^n$) converge exponentially fast
  • Harmonic sequences ($a_n = \frac{1}{n}$) converge slowly

Introduction to Cauchy Sequences

• Cauchy sequences are fundamental in analysis and the construction of real numbers
• Cauchy criterion: $\forall \varepsilon > 0, \exists N \in \mathbb{N}$ such that $|a_n - a_m| < \varepsilon$ for all $n, m \geq N$
  • Terms become arbitrarily close to each other as $n$ and $m$ increase
• Cauchy sequences are named after Augustin-Louis Cauchy, a French mathematician
• Cauchy sequences are defined in metric spaces, generalizing the concept from real numbers
  • Metric space: set with a distance function (metric) satisfying certain properties
• Cauchy sequences are a tool for proving convergence without knowing the limit

Relationship Between Cauchy and Convergent Sequences

• In complete metric spaces (including $\mathbb{R}$), Cauchy sequences are convergent
  • Completeness: every Cauchy sequence converges to a point within the space
• Convergent sequences are always Cauchy
  • If $\lim_{n \to \infty} a_n = L$, then $\{a_n\}$ is Cauchy
• In incomplete metric spaces, Cauchy sequences may not converge (within the space)
  • Example: $\mathbb{Q}$ is incomplete, sequence $\{1, 1.4, 1.41, 1.414, \ldots\}$ is Cauchy but converges to $\sqrt{2} \notin \mathbb{Q}$
• Cauchy sequences are used to construct real numbers from rational numbers
  • Dedekind cuts and Cauchy sequences provide equivalent constructions of $\mathbb{R}$

Examples and Applications

• Geometric sequences ($a_n = ar^n$) are monotone and convergent for $|r| < 1$
  • Increasing if $0 < r < 1$, decreasing if $-1 < r < 0$
  • $\lim_{n \to \infty} ar^n = 0$ for $|r| < 1$
• Harmonic sequence ($a_n = \frac{1}{n}$) is decreasing and convergent
  • Bounded below by 0, $\lim_{n \to \infty} \frac{1}{n} = 0$
• Alternating harmonic sequence ($a_n = \frac{(-1)^{n+1}}{n}$) is not monotone but is Cauchy
  • Convergent to $\ln(2)$, can be shown using the alternating series test
• Monotone sequences appear in optimization problems and numerical methods
  • Gradient descent, Newton's method, and fixed-point iteration generate monotone sequences
• Cauchy sequences are used in the construction of real numbers and in functional analysis
  • Lp spaces, Banach spaces, and Hilbert spaces are complete metric spaces

Common Pitfalls and Misconceptions

• Not all bounded sequences are convergent (oscillating sequences)
  • Example: $a_n = (-1)^n$ is bounded but not convergent
• Not all convergent sequences are monotone
  • Example: $a_n = \frac{(-1)^n}{n}$ converges to 0 but alternates between positive and negative
• Monotonicity does not imply strict monotonicity
  • Constant sequences are both increasing and decreasing
• Cauchy sequences are not always easy to identify
  • May require clever manipulations or estimates to prove Cauchy criterion
• Incomplete metric spaces have Cauchy sequences that do not converge within the space
  • Convergence in $\mathbb{R}$ does not imply convergence in $\mathbb{Q}$

Practice Problems and Exercises

1. Prove that the sequence $a_n = \frac{n}{n+1}$ is increasing and find its limit.
2. Show that the sequence $a_n = \frac{1}{2^n}$ is decreasing and converges to 0.
3. Determine whether the sequence $a_n = \cos(\frac{\pi n}{2})$ is monotone. Is it convergent?
4. Prove that the sequence $a_n = \sqrt{n}$ is unbounded and not Cauchy.
5. Show that the sequence $a_n = \frac{1}{n^2}$ is Cauchy using the definition of Cauchy sequences.
6. Find an example of a sequence that is Cauchy but not monotone.
7. Construct a monotone sequence that converges to $\sqrt{3}$.
8. Prove that if $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences, then $\{a_n + b_n\}$ is also Cauchy.