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Learn/what is a proof/How to Write Your First Proof

Lesson subsection

How to Write Your First Proof

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

Writing your first proofs feels awkward because you are learning a new language: mathematical English combined with logical structure.

A good way to start:

  1. Restate the goal in your own words.
  2. List what you know: assumptions, definitions, previously proved results.
  3. Try to connect what you know to what you need, starting from the assumptions.
  4. Write in clear sentences, not just algebra. Explain why each step is valid.

It's normal if early proofs feel clumsy. Clarity and correctness matter more than style. Style improves automatically with practice.

TL;DR — key idea

Start by restating the goal, listing your assumptions, and connecting them step by step. Focus on correctness and clarity; elegance comes later.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

  • 1

    Outline (in bullets) the steps you would take before writing a full proof of a simple statement, like 'the sum of two even integers is even'.

  • 2

    Explain why it is important to write sentences and not just chains of symbols when you are learning to prove things.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.