Core proof track
Mathematical Induction
Work through these bite-sized lessons. Each subsection has reading, notebook prompts, and an optional AI critique to refine your understanding.
Recommended: move in order, but you can jump around if you’re reviewing specific ideas.
Subsections
Click into a subsection to read, practice, and get feedback on your explanation.
The Logic Behind Induction
Induction proves a statement for all integers starting at n₀ by proving a base case and a step that pushes truth from n to n + 1.
Weak vs. Strong Induction
Weak induction assumes P(k) to prove P(k + 1); strong induction assumes all previous cases up to k. They are equivalent, but strong is often more convenient.
Induction for Algebraic Identities
For algebraic identities, induction usually assumes the formula up to n = k and then adds the next term, simplifying to the desired expression for k + 1.
Induction on Inequalities
Induction on inequalities uses the hypothesis to bound part of the expression for n + 1 and then shows that this bound is strong enough.
Induction for Recurrence Relations
Recurrence relation proofs usually pair a guessed formula with induction, often using strong induction when the next term depends on multiple previous ones.
Common Errors & Pitfalls
Most induction errors come from missing base cases, misusing the induction hypothesis, or hiding key steps. Clear structure and explicit hypotheses fix most of them.