A direct proof starts from the stated assumptions and uses definitions, known results and valid deductions until the desired conclusion follows.

For a statement

$$P\Rightarrow Q,$$

the direct strategy is

assume P
use definitions and established facts
...
therefore Q

A proof should make every nontrivial step justifiable; examples can suggest a theorem but do not prove a universal claim.

Counterexamples

A universal statement

$$\forall x\in S,\ P(x)$$

is disproved by one counterexample: an element $a\in S$ for which $P(a)$ is false.

This creates an important asymmetry:

• proving a universal claim requires an argument covering every allowed case;
• disproving it may require only one valid case.

Definitions are often the starting point of both proofs and counterexamples: before manipulating symbols, identify exactly what the claim means.