Monotone Convergence Theorem
Bounded monotone sequences converge.
Key idea
Bounded monotone sequences converge.
In depth
A sequence that is increasing and bounded above converges (to its supremum); a decreasing sequence bounded below converges to its infimum. It lets you prove convergence without knowing the limit.
Worked example
aₙ₊₁ = √(2 + aₙ), a₁ = √2, is increasing and bounded by 2, so converges (to 2).
Real-world uses
- Guaranteeing that a steadily growing balance or estimate settles at a limit
Common mistake
Forgetting to check boundedness.
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 Monotone Convergence Theorem?
Bounded monotone sequences converge. It is a topic at the expert level in Analysis & Topology.
Where is Monotone Convergence Theorem used in real life?
Real-world examples: Guaranteeing that a steadily growing balance or estimate settles at a limit.
What should I learn before Monotone Convergence Theorem?
Make sure you are comfortable with Limit Theorems for Sequences, the previous topic in Analysis & Topology.
What is a common mistake with Monotone Convergence Theorem?
Forgetting to check boundedness.
How can I practise Monotone Convergence Theorem?
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 Monotone Convergence Theorem?
Move on to Subsequences 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.