Lesson subsection
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.
Best flow: read → think on paper → write a short explanation → refine with feedback.
Examples are useful for building intuition: they help you guess which statements might be true. But examples alone can never guarantee a statement is always true.
For a universal statement like "for all integers n, P(n) holds," you could check a million values of n and still miss a counterexample. A proof, on the other hand, shows that no counterexample can exist.
A powerful habit in problem solving is:
TL;DR — key idea
Examples suggest truth; proofs guarantee it. Examples can show a statement is false, but only a proof can show it is always true.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Think of a statement about integers that you might try to check by plugging in values. Describe why examples alone are not enough.
Invent a false 'pattern' that looks true for the first few cases but eventually fails. Explain how a proof would reveal the failure.
Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.
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.