Real Analysis

Cauchy criterion of convergence

Q: FAQ 1: Why use the Cauchy criterion when we can find the limit directly?

In many advanced problems, finding the explicit limit $L$ is extremely difficult or impossible. The Cauchy criterion allows us to prove convergence purely by examining the internal behavior of sequence terms.

Q: FAQ 2: What is the single most common reason candidates lose marks on sequence questions?

Confusing the condition $a_{n+1} - a_n \to 0$ with the Cauchy criterion. Terms approaching zero is necessary but grossly insufficient for Cauchy or convergence.

Q: FAQ 3: How can candidates solve Cauchy and convergence questions?

Look for known standard benchmarks (p-series, geometric tails, monotonicity with boundedness) and immediately rule out unbounded or oscillating options.

The Cauchy criterion states that a sequence of real numbers converges if and only if it is a Cauchy sequence, providing an internal test for convergence without needing to know the actual limit.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Cauchy's Convergence Criterion for Sequences

A sequence of real numbers $\langle a_n \rangle$ is convergent in $\mathbb{R}$ if and only if for every $\varepsilon > 0$, there exists a positive integer $N$ such that $|a_m - a_n| < \varepsilon$ for all $m, n \ge N$.

📐 Geometric Interpretation

Geometrically, as we move far out along the sequence, the terms cluster arbitrarily close to one another, forming an increasingly tight band of diameter less than any preassigned $\varepsilon > 0$.

⚙️ Algebraic & Structural Meaning

It translates the topological property of convergence (which requires knowledge of the limit $L$) into an intrinsic algebraic condition involving only the terms of the sequence themselves.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Completeness of the underlying space ($\mathbb{R}$)	`Metric space $\mathbb{R}$`	Strictly Mandatory / Necessary	In incomplete spaces like $\mathbb{Q}$, Cauchy sequences (e.g., decimal approximations of $\sqrt{2}$) fail to converge within the space.
Tail index threshold ($m, n \ge N$)	`Discrete set of indices $\mathbb{N}$`	Strictly Mandatory	Failing to hold for all sufficiently large indices invalidates the tail contraction required for convergence.
Arbitrariness of $\varepsilon > 0$	`Open ray $(0, \infty)$`	Essential for conclusion	If the condition holds only for a fixed $\varepsilon$, the sequence need not converge (it may oscillate boundedly).

Useful results and shortcuts

Use these identities and results when their conditions apply:

Ratio Test for Sequence Tail Bounds

$$ |a_{n+1} - a_n| \le r^n |a_1 - a_0|, \quad 0 < r < 1 $$

Rule: If the consecutive difference satisfies a geometric bound with common ratio $r < 1$, the sequence is automatically Cauchy and convergent.

Application: Directly applied in UP PGT to determine convergence of recursively defined sequences like $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$.

Bounded Monotonic Convergence Equivalence

$$ \text{Monotonic} + \text{Bounded} \iff \text{Cauchy} \iff \text{Convergent in } \mathbb{R} $$

Rule: For monotonic sequences, never test Cauchy directly; verify boundedness to instantly establish convergence.

Alternating Harmonic & p-Series Tail Estimates

$$ |s_m - s_n| = \left| \sum_{k=n+1}^{m} \frac{1}{k^p} \right| < \varepsilon $$

Rule: Use integral comparison test on tails $\int_{n}^{m} \frac{1}{x^p} dx$ to verify Cauchy property for series.

Application: Frequently asked in DSSSB PGT for series convergence testing.

Subsequence Isolation Shortcut

$$ \text{If } \langle a_n \rangle \text{ is Cauchy and has a convergent subsequence } a_{n_k} \to L, \text{ then } a_n \to L $$

Rule: If a Cauchy sequence has even one convergent subsequence, the entire sequence converges to that exact limit.

Application: Used to eliminate distractor options in multiple-choice assertion questions.

Cauchy Condensation Test Link

$$ \sum a_n \text{ converges } \iff \sum 2^k a_{2^k} \text{ converges} $$

Rule: Transform heavy monotonic decreasing series into dyadic blocks to test Cauchy criterion instantly.

Rapid Elimination via Unboundedness

$$ \text{If } \lim_{n \to \infty} a_n = \infty \text{ or oscillates finitely without bound}, \text{ discard Cauchy} $$

Rule: Any Cauchy sequence in $\mathbb{R}$ is bounded. Instantly eliminate any option claiming a divergent or unbounded sequence is Cauchy.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Assuming Cauchy in Rational Numbers ($\mathbb{Q}$)

Common Pitfall: Candidates assume Cauchy sequences always converge to a number within the same set, forgetting completeness.

Counterexample: $a_n = \left(1 + \frac{1}{n}\right)^n \text{ or decimal truncations of } \sqrt{2} \text{ in } \mathbb{Q}$

Why it fails: The sequence is Cauchy, but its limit $\sqrt{2}$ or $e$ does not belong to $\mathbb{Q}$, so it diverges within $\mathbb{Q}$.

🛡️ Exam Defense: Always check the underlying metric space domain ($\mathbb{R}$ vs. $\mathbb{Q}$). Real analysis PGT exams explicitly exploit $\mathbb{Q}$ incompleteness.

⚠️ COMMON ERROR #02

Confusing 'Infinitely Close' with 'Convergent'

Common Pitfall: Assuming that $\lim_{n \to \infty} (a_{n+1} - a_n) = 0$ implies the sequence is Cauchy or convergent.

Counterexample: $a_n = \sqrt{n}$

Why it fails: Here $a_{n+1} - a_n \to 0$ as $n \to \infty$, but the sequence is unbounded and diverges to infinity.

🛡️ Exam Defense: Remember that $a_{n+1} - a_n \to 0$ is a necessary condition for convergence of certain series/sequences, but NOT sufficient for Cauchy criterion.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Cauchy Sequence Criterion	Metric space completeness, arbitrary $\varepsilon$	$\|a_m - a_n\| < \varepsilon$ for $m,n \ge N$	Tail terms cluster within an arbitrarily small neighborhood	Words: 'Cauchy', 'completeness', 'no limit known'
Bolzano-Weierstrass Theorem	Bounded sequence in $\mathbb{R}$	$\exists$ convergent subsequence	Infinitely many points cluster around at least one accumulation point	Words: 'Bounded', 'subsequence', 'limit point'
Monotone Convergence Theorem	Monotone sequence + Bounded	$\lim a_n = \sup \{a_n\}$ or $\inf \{a_n\}$	Sequence moves monotonically and hits a rigid ceiling/floor	Words: 'Monotonic', 'bounded', recursive formulas
Cantor's Intersection Theorem	Nested closed bounded intervals in $\mathbb{R}$	$\bigcap I_n \neq \emptyset$	Sequence of shrinking closed segments traps a unique real number	Words: 'Nested intervals', 'compactness'

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Determine whether the sequence defined by $a_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ is a Cauchy sequence in $\mathbb{R}$.

(A) Yes, because $a_{n+1} - a_n \to 0$(B) No, because it is divergent (Harmonic sequence)(C) Yes, because it is bounded(D) Insufficient data

📚 Method A: Formal Textbook Derivation

Evaluate $|a_{2n} - a_n| = \frac{1}{n+1} + \dots + \frac{1}{2n} \ge n \cdot \frac{1}{2n} = \frac{1}{2}$. Choosing $\varepsilon = \frac{1}{3}$ violates the Cauchy condition for arbitrarily large $n$. Thus, it is not Cauchy.

⚡ Method B: 25-Second Exam Shortcut

Recognize $a_n$ as the partial sum of the harmonic series. Harmonic series diverges, so its partial sum sequence is not Cauchy.

💡 Tip: Never confuse the term vanishing ($\frac{1}{n} \to 0$) with Cauchy property of partial sums.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If a sequence $\langle a_n \n \rangle$ satisfies $|a_{n+1} - a_n| \leq \frac{1}{2^n}$ for all $n \in \mathbb{N}$, then the sequence is:

(A) Divergent(B) Cauchy and convergent(C) Bounded but not Cauchy(D) Oscillatory

📚 Method A: Formal Textbook Derivation

Using triangle inequality for $m > n$: $|a_m - a_n| \le \sum_{k=n}^{m-1} |a_{k+1} - a_k| \le \sum_{k=n}^{m-1} \frac{1}{2^k} < \frac{1}{2^{n-1}} < \varepsilon$ for sufficiently large $N$. Hence it is Cauchy and complete in $\mathbb{R}$ implies convergence.

⚡ Method B: 25-Second Exam Shortcut

Geometric bound sum $\sum \frac{1}{2^n} < \infty$ immediately guarantees Cauchy property.

💡 Tip: Any sequence with exponential consecutive difference bound is strictly Cauchy.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: Every Cauchy sequence of real numbers is: $...$

A Bounded

B Monotonic

C Strictly increasing

D Convergent in $\mathbb{Q}$

Derivation & Analysis:

Every Cauchy sequence in a metric space is bounded. Monotonicity is not guaranteed.

MCQ 2: Which of the following sequences is Cauchy in $\mathbb{R}$? $...$

A $a_n = \sum_{k=1}^n \frac{1}{k}$

B $a_n = (-1)^n n$

C $a_n = \sin(n)$

D $a_n = \sum_{k=1}^n \frac{1}{k^2}$

Derivation & Analysis:

The partial sums of $\sum \frac{1}{k^2}$ converge, hence form a Cauchy sequence. Harmonic diverges, $(-1)^n n$ and $\sin(n)$ do not form Cauchy sequences.

MCQ 3: If $\langle a_n \rangle$ is a Cauchy sequence, then the sequence $\langle |a_n| \rangle$ is: $...$

A Not Cauchy

B Cauchy

C Convergent to zero

D Unbounded

Derivation & Analysis:

By the reverse triangle inequality ||a_m| - |a_n|| \le |a_m - a_n| < \varepsilon, showing $\langle |a_n| \rangle$ is also Cauchy.

MCQ 4: In the set of rational numbers $\mathbb{Q}$, the sequence $a_n = \left(1 + \frac{1}{n}\right)^n$ is: $...$

A Cauchy but not convergent in $\mathbb{Q}$

B Convergent in $\mathbb{Q}$

C Not Cauchy

D Divergent to infinity

Derivation & Analysis:

The sequence is Cauchy in $\mathbb{Q}$ and $\mathbb{R}$, but its limit $e$ is irrational, so it is not convergent within $\mathbb{Q}$.

MCQ 5: The necessary and sufficient condition for a sequence of real numbers to converge is that it be: $...$

A Bounded

B Monotonic

C Cauchy

D Arithmetic progression

Derivation & Analysis:

By Cauchy's General Principle of Convergence, a real sequence converges if and only if it is Cauchy.

MCQ 6: If $\langle a_n \rangle$ and $\langle b_n \rangle$ are Cauchy sequences in $\mathbb{R}$, then $\langle a_n + b_n \rangle$ is: $...$

A Not Cauchy

B Cauchy

C Always zero

D Unbounded

Derivation & Analysis:

Linear combinations of Cauchy sequences are Cauchy due to the triangle inequality.

MCQ 7: If $a_n = \frac{\sin n}{n}$, then the sequence $\langle a_n \rangle$ is: $...$

A Not Cauchy

B Cauchy and converges to 0

C Divergent

D Unbounded

Derivation & Analysis:

Since $|\sin n| \le 1$, $\frac{\sin n}{n} \to 0$, and bounded monotonic/null sequence properties confirm it is Cauchy.

MCQ 8: Assertion (A): Every convergent sequence is Cauchy. Reason (R): Every Cauchy sequence in $\mathbb{R}$ is convergent. $...$

A Both A and R are true and R is the correct explanation of A

B Both A and R are true but R is not the correct explanation of A

C A is true but R is false

D A is false but R is true

Derivation & Analysis:

Both statements are fundamental theorems of real analysis, representing the equivalence in complete spaces.

MCQ 9: If a sequence satisfies $|a_{n+2} - a_{n+1}| \le k |a_{n+1} - a_n|$ with $0 < k < 1$, the sequence is: $...$

A Cauchy

B Divergent

C Oscillatory

D Constant

Derivation & Analysis:

This is a contraction sequence condition, which guarantees it is a Cauchy sequence.

Q10

MCQ 10: Let $a_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!}$. The sequence $\langle a_n \rangle$ is: $...$

A Cauchy

B Divergent

C Unbounded

D Oscillatory

Derivation & Analysis:

This sequence converges to $e$, hence it is Cauchy.

Common questions

In many advanced problems, finding the explicit limit $L$ is extremely difficult or impossible. The Cauchy criterion allows us to prove convergence purely by examining the internal behavior of sequence terms.

Confusing the condition $a_{n+1} - a_n \to 0$ with the Cauchy criterion. Terms approaching zero is necessary but grossly insufficient for Cauchy or convergence.

Look for known standard benchmarks (p-series, geometric tails, monotonicity with boundedness) and immediately rule out unbounded or oscillating options.