Direct Proof Examples in Math

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

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

Concept Recap

A direct proof establishes a statement $P \Rightarrow Q$ by assuming $P$ is true and using logical steps, definitions, and known theorems to arrive at $Q$ — the most straightforward proof strategy.

Start from what you know (the hypotheses) and chain logical steps forward until you reach what you want to prove — no detours, no tricks, just forward reasoning.

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: A direct proof assumes the hypothesis $P$ is true and chains definitions, algebra, and known theorems forward until it reaches the conclusion $Q$ .

Common stuck point: The procedure for direct proof is the easy part; the trap is secretly assuming the conclusion $Q$ and reasoning toward the hypothesis. Asking "Can I start from the hypothesis and reach the conclusion using only forward steps, never assuming the conclusion is false?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I start from the hypothesis and reach the conclusion using only forward steps, never assuming the conclusion is false?

Worked Examples

Example 1

easy

Prove directly: the sum of an even integer and an odd integer is odd.

Answer

a+b = 2(m+n)+1 \text{ is odd} \quad \square

First step

Let

a

be even and

b

be odd. By definition,

a = 2m

and

b = 2n+1

for integers

m, n

Full solution

2
Then $a + b = 2m + 2n + 1 = 2(m+n) + 1$ .
3
Since $m+n$ is an integer, $a+b = 2(m+n)+1$ is odd by definition.

A direct proof proceeds from the hypothesis (definitions of even and odd) to the conclusion (the sum is odd) by explicit algebraic steps. The form

2k+1

for some integer

k

is the definition of odd.

Example 2

medium

Prove directly: for all real

x

(x-1)(x+1) = x^2 - 1

Example 3

medium

Prove directly: if

n

is an integer then

n^2 + 3n + 2

is even.

Example 4

medium

Prove directly: if

n

is an integer, then

4n^2 + 4n + 1

is odd.

Example 5

hard

Prove directly: for all positive reals

a, b

a^2 + b^2 \ge 2ab

Practice Problems

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

Example 1

easy

Prove directly: if

x > y > 0

, then

x^2 > y^2

Example 2

medium

Prove directly: for all integers

n

n^2 + n

is even.

Example 3

easy

In a direct proof of 'if

n

is even then

n^2

is even', what do you assume FIRST?

Example 4

easy

Complete: 'To prove

n

even

\Rightarrow n^2

even, start

n=2k

, so

n^2 =

___.'

Example 5

easy

In a direct proof of 'if

n

is odd then

n+1

is even', what is the conclusion you must REACH?

Example 6

easy

Direct proof that the sum of two even integers is even: assume

a=2j

and

b=2k

. What is

a+b

Example 7

easy

Direct proof that

3 \mid n \Rightarrow 3 \mid n^2

: assume

n=3k

. What is

n^2

Example 8

easy

Direct proof that 'if

x>2

then

x^2>4

': from

x>2

and

x>0

, what can you multiply to conclude?

Example 9

easy

Which opening is correct for a direct proof of

P\Rightarrow Q

: (A) 'Assume

Q

.' or (B) 'Assume

P

.'?

Example 10

easy

Direct proof that the product of two odd integers is odd: assume

a=2j+1

b=2k+1

. Expand

ab

Example 11

medium

Complete the direct proof that

n

odd

\Rightarrow n^2

odd: assume

n=2k+1

, then

n^2 =

? Show it is odd.

Example 12

medium

Direct proof: if

a \mid b

and

b \mid c

then

a \mid c

. Set up the definitions and finish.

Example 13

medium

Direct proof that if

n=2k+1

then

n^2-1

is divisible by

8

. Complete it.

Example 14

medium

Direct proof that the sum of an even and an odd integer is odd: assume

a=2j

b=2k+1

. Finish.

Example 15

medium

Direct proof that if

x

and

y

are rational then

x+y

is rational. Set up and conclude.

Example 16

medium

Direct proof: if

a

and

b

are consecutive integers then

a+b

is odd. Complete it.

Example 17

medium

Find the flaw in this 'direct proof' that

n^2

even

\Rightarrow n

even: 'Assume

n

is even,

n=2k

, then

n^2=4k^2

is even. Done.' What went wrong?

Example 18

medium

Direct proof that if

5 \mid n

and

5 \mid m

then

5 \mid (n+m)

. Set up and finish.

Example 19

medium

Direct proof that if

x

is rational and nonzero then

\frac{1}{x}

is rational. Complete it.

Example 20

challenge

Direct proof that if

n

is an integer then

n^3-n

is divisible by

6

. Complete the argument.

Example 21

challenge

Direct proof that the sum of the first

n

odd numbers is

n^2

, WITHOUT induction. Give a direct algebraic argument.

Example 22

challenge

Direct proof that if

a^2+b^2=c^2

with integers and

a,b

both odd, derive a contradiction-free direct statement about

c^2 \bmod 4

. Complete it.

Example 23

easy

In a direct proof of '

n

is a multiple of 4

\Rightarrow n

is even', what is the opening assumption?

Example 24

easy

Finish: 'If

n = 4k

, then

n = 2(2k)

, so

n

is ____.'

Example 25

easy

Direct proof: if

n

is even then

3n

is even. Set up and finish.

Example 26

easy

Direct proof: for all real

x

x^2 \ge 0

. Briefly justify.

Example 27

easy

Direct proof: if

n = 6k

, then

n

is divisible by 3. Finish in one line.

Example 28

medium

Direct proof: if

a, b

are integers with

a

even, then

ab

is even. Show the steps.

Example 29

medium

Direct proof: if

a \mid b

and

a \mid c

, then

a \mid (b+c)

. Set up and finish.

Example 30

medium

Direct proof: for all reals

a, b

(a+b)^2 \ge 4ab

. Hint: expand.

Example 31

medium

Direct proof: if

x

is rational and

y

is rational, then

xy

is rational. Complete it.

Example 32

medium

Direct proof that if

n

is odd, then

n^2 - 1

is divisible by 4. Complete it.

Example 33

medium

Direct proof: if

a > 0

then

a + 1/a \ge 2

. (Assume real

a

Example 34

medium

Direct proof: if

a

and

b

are integers with

a \equiv b \pmod{m}

, then

a^2 \equiv b^2 \pmod{m}

Example 35

hard

Direct proof: if

n

is a positive integer, then

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

. Argue without induction, using Gauss's pairing.

Example 36

hard

Direct proof: if

a

is rational and

b

is irrational, must

a + b

be irrational?

Example 37

hard

Direct proof: if

n

is an odd integer, then

n^2 \equiv 1 \pmod 8

Example 38

hard

Direct proof: if

f

is even and

g

is odd, then

f \cdot g

is odd as a function.

Example 39

hard

Direct proof: if

a, b

are integers and

ab

is odd, then both

a

and

b

are odd. (Hint: prove the contrapositive directly via cases.)

Example 40

hard

Direct proof: for any integers

a, b

|a + b| \le |a| + |b|

(triangle inequality on

\mathbb{Z}

Example 41

challenge

Direct proof: if

n

is an integer,

n^2 \equiv 0

1 \pmod 4

Example 42

challenge

Direct proof: for all positive reals

a, b, c

a^2 + b^2 + c^2 \ge ab + bc + ca

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Related Concepts

Logical Statement Conditional Statement Proof (Intuition)

Background Knowledge

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

logical statementconditionalproof intuition