7.1 Direct Proof

A direct proof begins with the assumptions of a statement and derives the conclusion by valid steps. It follows the natural direction of implication.

A typical direct proof has the form:

To prove it directly, assume $P$ and show $Q$.

This method is often the clearest proof technique because it does not change the goal. The proof follows the same direction as the statement.

For example, consider the claim:

If $n$ is even, then $n^2$ is even.

A direct proof starts by unpacking the assumption. If $n$ is even, then there exists an integer $k$ such that

Then

Since $2k^2$ is an integer, $n^2$ is even. The conclusion follows directly from the definition.

This example shows the standard rhythm of direct proof: expand definitions, perform valid transformations, and recognize the conclusion.

Direct proof works best when the conclusion is visibly contained in the assumptions. It is especially effective for statements involving definitions, algebraic identities, divisibility, order, inclusion, and simple implication.

Many direct proofs are short because the main work is choosing the right definition. To prove that one set is contained in another, take an arbitrary element of the first set and show it belongs to the second. To prove that a function is injective, assume two outputs are equal and show the inputs are equal. To prove that a sequence is bounded, produce a bound and verify it.

The structure is always the same: start from what is given, use definitions, and reach what is required.

A direct proof also makes dependencies visible. Each step should follow from an assumption, a definition, a known theorem, or an earlier step. If a step does not have such a source, the proof has a gap.

For example, to prove

one should write: let $x \in A$. Then show $x \in B$. Since $x$ was arbitrary, every element of $A$ belongs to $B$, so $A \subseteq B$.

This pattern avoids vague claims about sets and reduces inclusion to element-level reasoning.

For equality of sets, direct proof often proceeds by proving two inclusions:

means

and

Each inclusion can be proved directly.

Direct proof is also common in algebra. Suppose $f : G \to H$ is a group homomorphism. To prove that $f(e_G) = e_H$, begin with the homomorphism property:

Multiplying by the inverse of $f(e_G)$ in $H$ gives

The result follows by direct manipulation inside the group.

A direct proof should avoid unnecessary detours. It should not introduce contradiction, cases, or induction unless the statement requires them. The goal is to move from assumptions to conclusion with the shortest clear path.

The main danger is hiding too much in phrases such as “clearly” or “obvious.” Direct proof still requires justification. If a step depends on a definition, state the definition. If it depends on a theorem, name or cite the theorem. If it depends on algebra, show enough of the calculation for the reader to verify it.

A useful template is:

Assume the hypotheses. Unpack the definitions. Apply known facts. Transform the expression or object. Arrive at the required conclusion.

For example, to prove that the sum of two even integers is even:

Assume $a$ and $b$ are even. Then there exist integers $m$ and $n$ such that

Then

Since $m+n$ is an integer, $a+b$ is even.

The proof is direct because it starts with the definition of evenness and ends with the same definition applied to $a+b$.

Direct proof is the default method. Before trying a more elaborate technique, ask whether the conclusion follows by expanding definitions and applying standard rules. Many theorems do.