Practice Proof Techniques in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

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

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

Showing a random 20 of 50 problems.

Example 1

hard
Identify a flaw in this proof: 'For all n≥1n\ge 1, all groups of nn horses have the same color. Base n=1n=1: trivial. Step: assume true for nn; for n+1n+1 horses, drop the last, the first nn are same colour; drop the first, the last nn same colour; together all n+1n+1 same.'

Example 2

hard
Prove that there exist irrational a,ba,b with aba^b rational. (Use a non-constructive case split.)

Example 3

challenge
Use the well-ordering principle to prove the division algorithm: for a∈Za\in\mathbb Z, b∈Nb\in\mathbb N, there exist unique q,rq,r with a=bq+ra=bq+r and 0≤r<b0\le r<b.

Example 4

medium
Claim: 'If abab is odd then both aa and bb are odd.' Which technique avoids messy casework, and what is its core?

Example 5

easy
Name four proof techniques, give a one-sentence description of each, and identify which is best suited to prove: 'For all n≥1n \ge 1, 3∣(n3−n)3 \mid (n^3 - n).'

Example 6

hard
Which technique should you use to prove: 'For every ϵ>0\epsilon>0, there exists δ>0\delta>0 such that ∣x−2∣<δ|x-2|<\delta implies ∣x2−4∣<ϵ|x^2-4|<\epsilon.'?

Example 7

medium
Claim: 'Every integer n≥2n\ge2 has a prime factor.' Which technique fits, and what is its structural seed?

Example 8

medium
Prove using mathematical induction: 3n>2n+13^n > 2n+1 for all n≥2n \ge 2.

Example 9

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

Example 10

challenge
Claim: 'Among any 55 points in a unit square, two are within 22\frac{\sqrt2}{2} of each other.' Which technique proves it, and what is the partition?

Example 11

medium
Prove by contrapositive: if n2n^2 is even, then nn is even.

Example 12

medium
Existence proof: show ∃x∈R\exists x\in\mathbb R with x3=2x^3=2.

Example 13

hard
Use strong induction to prove every integer n≥2n\ge 2 is a product of primes.

Example 14

hard
The pigeonhole principle is a proof technique. State it in one sentence and give one application: among 1313 people, at least two share what?

Example 15

easy
What is the base case typically in a proof by induction on positive integers?

Example 16

easy
Identify the technique: 'To prove ∃x:P(x)\exists x: P(x), exhibit a specific x0x_0 with P(x0)P(x_0).'

Example 17

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

Example 18

medium
Prove by induction that 1+2+⋯+n=n(n+1)21+2+\cdots+n=\frac{n(n+1)}{2} for all n≥1n\ge 1.

Example 19

hard
To prove a function ff is injective, the standard direct approach is to assume what and conclude what?

Example 20

medium
Claim: 'The equation x2+1=0x^2+1=0 has no real solution.' Which technique fits, and what is the core fact?
← Back to Proof Techniques Worked Examples