# Direct proof

Start from the assumptions and move step by step, using definitions and known facts, until you reach the conclusion.

Maths (university) · Part 01: Logic, sets and proof · Card 5 of 9 · Last updated 2026-10-19

## Analogy

Following a recipe: each line follows from the one before, ending at the finished dish.

## Example

**Claim: n even gives n^2 even**

n even means n = 2k for a whole k.
n^2 = (2k)^2 = 4k^2.
4k^2 = 2(2k^2).
That is 2 times a whole number, so even.

## In practice

Direct proof is the first thing to try. Unpack the definitions in the assumption, then see where algebra leads.

## Practice

### Flashcards

- **Write an even number n using a whole number k.** n = 2k.

## Questions people ask

### How do I start a direct proof?

Write down what you assume and what you must show. Then replace words with their definitions. 'n is even' becomes 'n = 2k for some whole number k'. Work forward with algebra until the expression matches the definition of the conclusion. Finish by stating that you have reached it.

### Does checking examples count as a proof?

No. Examples can suggest a pattern, but a proof must cover every case. Checking n = 2, 4 and 6 does not prove 'n even gives n squared even' for all n. You need an argument with a general n, like writing n = 2k.

## Related concepts

- [Contrapositive](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-3)
- [Proof by contradiction](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-6)
- [Induction](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-7)
- [Epsilon-delta definition](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-9)

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