MATH-18 · Advanced · ENGLISH · PLANNED
Build the picture, explain the rule, work an example, then justify it.
Study time is an initial activity estimate, excluding extra reading, revision and extended laboratory or project work. Videos and question bodies have not been produced by adding this outline.
Complete these courses or demonstrate equivalent knowledge. These are recommended prerequisites, not enforced enrolment locks.
YouTube playlist: ARK Mathematics | Real Analysis and Metric Spaces
Playlist title reserved in the plan; no YouTube playlist or upload has been created yet.
Sign in to save this learning pathPractical outcome: Prove convergence claims used in a numerical algorithm.
Explain ordered fields using a visual and one worked real-world example. Context: Prove convergence claims used in a numerical algorithm. End with a prediction or check-for-understanding.
Explain completeness using a visual and one worked real-world example. Context: Prove convergence claims used in a numerical algorithm. End with a prediction or check-for-understanding.
Explain supremum using a visual and one worked real-world example. Context: Prove convergence claims used in a numerical algorithm. End with a prediction or check-for-understanding.
Explain sequence convergence proofs using a visual and one worked real-world example. Context: Prove convergence claims used in a numerical algorithm. End with a prediction or check-for-understanding.
Use the first concepts from this section. Prove convergence claims used in a numerical algorithm. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain monotone convergence using a visual and one worked real-world example. Context: Prove convergence claims used in a numerical algorithm. End with a prediction or check-for-understanding.
Explain cauchy sequences using a visual and one worked real-world example. Context: Prove convergence claims used in a numerical algorithm. End with a prediction or check-for-understanding.
Explain subsequences using a visual and one worked real-world example. Context: Prove convergence claims used in a numerical algorithm. End with a prediction or check-for-understanding.
Explain bolzano-weierstrass theorem using a visual and one worked real-world example. Context: Prove convergence claims used in a numerical algorithm. End with a prediction or check-for-understanding.
Prove convergence claims used in a numerical algorithm. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Ordered fields, Completeness, Supremum and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Explain when a numerical minimum is guaranteed to exist.
Explain metric spaces using a visual and one worked real-world example. Context: Explain when a numerical minimum is guaranteed to exist. End with a prediction or check-for-understanding.
Explain open and closed sets using a visual and one worked real-world example. Context: Explain when a numerical minimum is guaranteed to exist. End with a prediction or check-for-understanding.
Explain compactness using a visual and one worked real-world example. Context: Explain when a numerical minimum is guaranteed to exist. End with a prediction or check-for-understanding.
Explain connectedness using a visual and one worked real-world example. Context: Explain when a numerical minimum is guaranteed to exist. End with a prediction or check-for-understanding.
Use the first concepts from this section. Explain when a numerical minimum is guaranteed to exist. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain sequential continuity using a visual and one worked real-world example. Context: Explain when a numerical minimum is guaranteed to exist. End with a prediction or check-for-understanding.
Explain uniform continuity using a visual and one worked real-world example. Context: Explain when a numerical minimum is guaranteed to exist. End with a prediction or check-for-understanding.
Explain extreme value theorem using a visual and one worked real-world example. Context: Explain when a numerical minimum is guaranteed to exist. End with a prediction or check-for-understanding.
Explain contraction mapping theorem using a visual and one worked real-world example. Context: Explain when a numerical minimum is guaranteed to exist. End with a prediction or check-for-understanding.
Explain when a numerical minimum is guaranteed to exist. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Metric spaces, Open and closed sets, Compactness and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Check whether a limiting operation is justified in a model.
Explain derivative existence using a visual and one worked real-world example. Context: Check whether a limiting operation is justified in a model. End with a prediction or check-for-understanding.
Explain mean value theorem proof using a visual and one worked real-world example. Context: Check whether a limiting operation is justified in a model. End with a prediction or check-for-understanding.
Explain riemann integrability using a visual and one worked real-world example. Context: Check whether a limiting operation is justified in a model. End with a prediction or check-for-understanding.
Explain pointwise convergence using a visual and one worked real-world example. Context: Check whether a limiting operation is justified in a model. End with a prediction or check-for-understanding.
Use the first concepts from this section. Check whether a limiting operation is justified in a model. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain uniform convergence using a visual and one worked real-world example. Context: Check whether a limiting operation is justified in a model. End with a prediction or check-for-understanding.
Explain interchanging limits and integrals using a visual and one worked real-world example. Context: Check whether a limiting operation is justified in a model. End with a prediction or check-for-understanding.
Explain equicontinuity using a visual and one worked real-world example. Context: Check whether a limiting operation is justified in a model. End with a prediction or check-for-understanding.
Explain arzela-ascoli theorem using a visual and one worked real-world example. Context: Check whether a limiting operation is justified in a model. End with a prediction or check-for-understanding.
Check whether a limiting operation is justified in a model. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Derivative existence, Mean value theorem proof, Riemann integrability and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Combine all three sections: Check whether a limiting operation is justified in a model. Submit an original worked artifact, evidence or tests, assumptions and a reflection. Rubric: correctness, method, practical usefulness, reproducibility and communication. Self-reviewed, not certified.
6–12 balanced questions sampling every section of Real Analysis and Metric Spaces, including interpretation, calculation or reasoning, a misconception and a new application. Passing is 70%; review errors before retrying.
Visual constructions, worked problems and proofs. Guided work leads to an independent attempt, with a diagram, explanation, trace, calculation or tested artifact. Question bodies are generated when opened in your private learning path. Programming submissions are stored as text; the platform does not execute submitted code.
Use original or appropriately licensed datasets. Practical science activities use models or supplied data unless a qualified supervisor has reviewed a physical experiment.
Open the interactive practice sampler →