Proof Techniques Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proof Techniques.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Proof techniques are standard strategies for establishing mathematical claims under different structures.

Choose the argument tool that matches the claim type and assumptions.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Proof techniques are the menu of standard strategies — direct, contradiction, contrapositive, induction, cases — and the skill is choosing the one that fits the statement you must establish.

Common stuck point: The procedure for proof techniques is the easy part; the trap is defaulting to one favorite technique for every problem. Asking "Have I matched the strategy to the claim's form before starting to write the proof?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Have I matched the strategy to the claim's form before starting to write the proof?

Worked Examples

Example 1

easy

Name four proof techniques, give a one-sentence description of each, and identify which is best suited to prove: 'For all

n \ge 1

3 \mid (n^3 - n)

Answer

n^3-n=n(n-1)(n+1);\text{ direct proof via consecutive integers works best}

First step

1. Direct proof: assume the hypothesis and derive the conclusion by logical steps.

Full solution

2
2. Proof by contradiction: assume the negation of the goal and derive a contradiction.
3
3. Proof by contrapositive: prove $\neg q \Rightarrow \neg p$ instead of $p \Rightarrow q$ .
4
4. Mathematical induction: prove a base case and an inductive step for statements indexed by $\mathbb{N}$ .
5
Best technique for $3 \mid (n^3-n)$ : direct proof. Factor: $n^3-n = n(n-1)(n+1)$ — three consecutive integers, so one is divisible by 3. Done.

Knowing multiple proof techniques and choosing the most efficient one for a given claim is a key mathematical skill. Factoring

n^3-n

reveals the consecutive-integer structure, making a direct proof immediate.

Example 2

medium

Compare direct proof and proof by contrapositive for: 'If

n^2

is even, then

n

is even.' Which technique is more natural here?

Example 3

medium

Prove by induction that

1+2+\cdots+n=\frac{n(n+1)}{2}

for all

n\ge 1

Example 4

medium

Prove by contradiction that

\sqrt{2}

is irrational (give the standard outline).

Example 5

medium

Prove by induction that

2^n>n

for all

n\ge 1

Example 6

hard

Use strong induction to prove every integer

n\ge 2

is a product of primes.

Example 7

hard

Prove that there exist irrational

a,b

with

a^b

rational. (Use a non-constructive case split.)

Example 8

hard

Prove by contradiction that there is no smallest positive rational number.

Example 9

challenge

Use the well-ordering principle to prove the division algorithm: for

a\in\mathbb Z

b\in\mathbb N

, there exist unique

q,r

with

a=bq+r

and

0\le r<b

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Which proof technique is most appropriate for: 'There exists a real number

x

such that

x^2 = 2

'? Apply it.

Example 2

medium

Prove using mathematical induction:

3^n > 2n+1

for all

n \ge 2

Example 3

easy

To prove 'if

n

is even then

n^2

is even', which technique most directly fits: direct proof, contradiction, or induction?

Example 4

easy

To prove '

\sqrt{2}

is irrational', which technique is standard?

Example 5

easy

To prove a statement holds 'for all positive integers

n

', which technique is the natural fit?

Example 6

easy

To disprove 'every prime is odd', which technique applies?

Example 7

easy

To prove 'if

n^2

is even then

n

is even', the direct route is awkward. Which technique simplifies it?

Example 8

easy

To prove '

|x|\ge 0

for all real

x

', which technique handles the sign of

x

best?

Example 9

easy

Identify the technique: 'Assume for contradiction there is a largest prime

p

...'. Which is being used?

Example 10

easy

Identify the technique: a proof shows 'base case

n=1

holds' then 'if it holds for

k

it holds for

k+1

'. Which is it?

Example 11

medium

Claim: 'For all integers

n

n^2+n

is even.' Which technique is cleanest, and what is the one-line core?

Example 12

medium

Claim: 'There is no smallest positive rational number.' Which technique fits, and what is the contradiction's seed?

Example 13

medium

Claim: '

1+2+\cdots+n=\frac{n(n+1)}{2}

.' Name the technique and state the inductive step's key equation.

Example 14

medium

Claim: 'If

ab

is odd then both

a

and

b

are odd.' Which technique avoids messy casework, and what is its core?

Example 15

medium

To disprove 'for all real

x

x^2 > x

', give the technique and a witness.

Example 16

medium

Claim: 'Every integer

n\ge2

has a prime factor.' Which technique fits, and what is its structural seed?

Example 17

medium

Identify the technique and the flaw: a proof of 'all horses are the same color' uses induction but its inductive step fails for

n=1\to2

. What technique, what gap?

Example 18

medium

Claim: 'If

n^2

is odd then

n

is odd.' Which technique is cleanest, and what is the equivalent statement you prove?

Example 19

medium

Claim: 'The equation

x^2+1=0

has no real solution.' Which technique fits, and what is the core fact?

Example 20

challenge

Claim: 'Among any

5

points in a unit square, two are within

\frac{\sqrt2}{2}

of each other.' Which technique proves it, and what is the partition?

Example 21

challenge

A claim is proved by checking

n=1,2,3,4

and asserting the pattern continues. Name the flawed 'technique' and the correct one, with the reason.

Example 22

challenge

Claim: '

\sqrt{2}+\sqrt{3}

is irrational.' Outline which technique and the key squaring move.

Example 23

easy

Which proof technique uses the contrapositive 'if not

Q

then not

P

' to prove 'if

P

then

Q

Example 24

easy

To disprove 'all swans are white', what type of evidence suffices?

Example 25

easy

In a proof by cases of a statement about

|x|

, what are the two natural cases?

Example 26

easy

To prove 'there are infinitely many primes', Euclid's argument is an example of which technique?

Example 27

medium

Prove directly: if

a

and

b

are even integers, then

a+b

is even.

Example 28

medium

Prove by contrapositive: if

n^2

is even, then

n

is even.

Example 29

medium

Prove by cases that

|xy|=|x||y|

for all reals.

Example 30

medium

Identify the technique used to prove uniqueness of an object satisfying property

P

: assume two such objects exist and derive that they are equal.

Example 31

hard

Identify a flaw in this proof: 'For all

n\ge 1

, all groups of

n

horses have the same color. Base

n=1

: trivial. Step: assume true for

n

; for

n+1

horses, drop the last, the first

n

are same colour; drop the first, the last

n

same colour; together all

n+1

same.'

Example 32

hard

Which technique should you use to prove: 'For every

\epsilon>0

, there exists

\delta>0

such that

|x-2|<\delta

implies

|x^2-4|<\epsilon

.'?

Example 33

hard

To prove a function

f

is injective, the standard direct approach is to assume what and conclude what?

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Related Concepts

Proof (Intuition)Contrapositive Quantifiers

Background Knowledge

These ideas may be useful before you work through the harder examples.

proof intuitioncontrapositivequantifiers