Cauchy Sequences
Terms get arbitrarily close to each other.
Key idea
Terms get arbitrarily close to each other.
In depth
A sequence is Cauchy if for every ε there is N such that |aₙ − aₘ| < ε for all n, m > N: the terms get arbitrarily close to each other. In the reals, Cauchy equals convergent, a form of completeness that needs no knowledge of the limit.
Worked example
Partial sums of Σ1/n² form a Cauchy sequence, so the series converges.
Formula
Real-world uses
- Checking convergence without knowing the limit, as used in iterative computer algorithms
Common mistake
Checking only consecutive differences |aₙ₊₁ − aₙ| → 0, which is not sufficient (harmonic series).
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Cauchy Sequences?
Terms get arbitrarily close to each other. It is a topic at the expert level in Analysis & Topology.
What is the formula for Cauchy Sequences?
The key formula is |aₘ − aₙ| < ε.
Where is Cauchy Sequences used in real life?
Real-world examples: Checking convergence without knowing the limit, as used in iterative computer algorithms.
What should I learn before Cauchy Sequences?
Make sure you are comfortable with Bolzano–Weierstrass Theorem, the previous topic in Analysis & Topology.
What is a common mistake with Cauchy Sequences?
Checking only consecutive differences |aₙ₊₁ − aₙ| → 0, which is not sufficient (harmonic series).
How can I practise Cauchy Sequences?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Cauchy Sequences?
Move on to Limsup and Liminf next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.