Core proof track
What is a Proof?
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.
What Mathematicians Mean by Proof
A proof is not just convincing; it is a logically complete argument that works in every valid case, not just in examples.
Examples vs. Proofs — Why They’re Different
Examples suggest truth; proofs guarantee it. Examples can show a statement is false, but only a proof can show it is always true.
Common Structures of Arguments
Proofs follow standard patterns like direct proof, contrapositive, contradiction, and casework. Learning these templates makes proofs far less intimidating.
How to Read a Proof (Actively)
Don’t just let the proof wash over you. Pause, question each step, track assumptions, and try to predict or reconstruct steps yourself.
How to Write Your First Proof
Start by restating the goal, listing your assumptions, and connecting them step by step. Focus on correctness and clarity; elegance comes later.
Common Beginner Mistakes
Beginner mistakes—like assuming the conclusion, only checking examples, or writing vaguely—are normal. Noticing them is the first step to fixing them.