Textbook
A proof is a chain of valid deductions from assumptions and established results. A universal assertion must be justified for every case covered by its conditions. An example illustrates one case; a single counterexample refutes a universal assertion.
A definition fixes meaning, an axiom is accepted in the chosen system, and a theorem is established by proof. A lemma is a proved assertion usually used in another result. These are roles in an exposition, not levels of reliability.
The expression $2x+1$ specifies a value, the equation $2x+1=7$ asks for an unknown, and the identity $(x+1)^2=x^2+2x+1$ holds for all permitted $x$. Testing one number does not turn an equality into an identity.
A proof begins by fixing its domain and assumptions. Each transition must follow from them or from an already justified step. For a universal claim, the chosen object remains arbitrary: no convenient extra property may quietly be assumed. A division step also needs grounds for its denominator being nonzero. A correct conclusion does not repair an invalid transition in the reasoning.
Definitions and further reading: Mathematics for Computer Science.
In plain language
A proof justifies a claim for an arbitrary admissible case rather than several successful examples. If $n$ is divisible by 4, then $n=4k=2(2k)$ for some integer $k$, so $n$ is even. Each transition rests on an assumption or definition, like a route whose turns all have a valid passage. Three working kettles help test a conjecture but cannot certify every kettle in the world; individual trials do not replace a general justification.
Example
The example moves from two numerical cases to an arbitrary integer $k$. The last step covers all integers divisible by 4, including zero and negative ones.
Proof: Why an Example Is Not Enough
Example
The number 4 is even, but this is just one case.
Step 1 of 3: Example
Read all steps
Example
The number 4 is even, but this is just one case.
- State
- 4current2 × 2current
Another example
Eight is also even; a finite check still is not a proof.
- State
- 8current2 × 4current
Arbitrary case
The integer 2k establishes divisibility by 2 for every allowed k.
- State
- n = 4kcurrentn = 2(2k)current2k ∈ ℤcurrent
Two details matter in the final line: $n=4k$ comes from the assumption, and $2k$ remains an integer. This licenses calling $n$ twice an integer, which means even. Positivity of $k$ was never used, so the proof retains zero and negative numbers. Checking only 4 and 8 would leave their shared reason for evenness unstated, however many successful examples were subsequently added.
Related concepts: Implication.