Proof as a Chain of Justified Steps

I want you to sit with this idea for a moment: a thousand examples cannot prove a claim in mathematics, but a single deductive proof can. Deductive proof works by starting from things we already know are true — definitions, postulates (rules accepted without proof), and theorems that have already been established — and moving forward in small, airtight steps until we reach the claim we set out to prove. Every step must be justified: you state what is true, then you name exactly why it must be true. No step floats on its own. Consider proving that vertical angles are congruent. We do not measure dozens of pairs and say 'they always look equal.' Instead, we reason from the Linear Pair Postulate — which states that two angles forming a linear pair are supplementary — and from the definition of supplementary angles. Those are our anchors. From them we derive, step by step, that both vertical angles equal the same quantity, so they must be equal to each other. The proof applies to every pair of vertical angles that will ever exist — not because we checked them all, but because the logic holds universally. Think of a proof as a chain: every link is a justified statement, and the chain is only as strong as its weakest link. If any step is unjustified — even if the conclusion feels obviously true — the proof is incomplete. That discipline is what gives mathematics its certainty. Hint: when you build a proof, ask yourself after each statement, 'What rule or fact allows me to write this?' The answer to that question is your justification.