How to Write Your First Proof

What a mathematical proof actually is

A proof is a finite sequence of statements where every statement is either an assumption you are permitted to make, a definition, a previously established result, or a logical consequence of earlier statements in the same argument. The last statement is the claim you set out to prove. Nothing in the chain may depend on a picture, on checking a few examples, or on the reader agreeing that something looks obvious.

This is why proofs feel strict. In everyday reasoning we accept a conclusion once we are persuaded. In mathematics, being persuaded is not enough; every step has to be defensible on its own. A useful test while writing is to ask of each sentence, if a skeptical reader stopped me here and asked why, could I point to a definition or an earlier line? If the answer is no, the step is a gap and the proof is incomplete.

The anatomy of a direct proof

Most claims you prove early on have the form: if P, then Q. A direct proof of such a statement follows a fixed skeleton. You assume that P is true, you unfold the definitions hiding inside P, you manipulate what you now know using algebra or logic, and you arrive at Q. You never assume Q; that is the thing you are trying to reach.

The reason this works is the meaning of implication. To show if P then Q, you only have to guarantee that Q holds in every case where P holds. So you deliberately step into the world where P is true and demonstrate that Q is forced. Writing the words Assume, Then, and Therefore at the right places keeps this flow visible to your reader.

• Assume the hypothesis P and name any objects it introduces.
• Replace defined terms with their definitions so you have equations or concrete conditions to work with.
• Reason forward using algebra, logic, or known theorems.
• Reach the conclusion Q and state that the implication is proved.

Definitions are your starting material

Beginners often stare at a claim without moving because they have not translated the words into usable conditions. Definitions are the bridge. If a claim mentions an even integer, the working definition is that an integer n is even when there exists an integer k with n equal to 2k. That existence statement is the concrete handle you actually compute with.

Get into the habit of writing definitions down explicitly at the top of a proof. Turning even n into n equals 2k for some integer k converts a vague noun into an equation. Almost every direct proof in number theory and logic gets its momentum from this one move: name the object, then expand its definition into symbols you can rearrange.

A worked example: the sum of two even numbers is even

Claim: if a and b are even integers, then a plus b is even. We prove it directly. Assume a and b are even. By the definition of even, there is an integer m with a equal to 2m, and there is an integer n with b equal to 2n. These two equations are everything the hypothesis gives us.

Now compute the quantity we care about. a plus b equals 2m plus 2n, which factors as 2 times the quantity m plus n. Because m and n are integers, their sum m plus n is also an integer; call it k. Then a plus b equals 2k where k is an integer. That is precisely the definition of an even number, so a plus b is even. The implication is proved.

Notice how mechanical the argument is once the definitions are unfolded. We assumed the hypothesis, wrote each even number as two times an integer, added, factored out the 2, and recognized the result as even. The same skeleton proves that the sum of two odd numbers is even.

Common mistakes beginners make

The most frequent error is assuming the conclusion. If you find yourself starting a proof with Suppose a plus b is even, you have quietly assumed the very thing you were asked to establish. Reread your first line and confirm you assumed the hypothesis, not the goal.

A second common slip is reusing the same variable for two different objects. If both a and b were even, writing a equals 2k and b equals 2k forces a and b to be equal, which is not given. Use distinct letters, such as m and n, for independently chosen integers. Finally, avoid proof by example. Checking that 4 plus 6 is even demonstrates one case; it does not prove the claim for all even integers, and a general claim demands a general argument.

• Do not assume the conclusion; assume only the hypothesis.
• Give distinct objects distinct variable names.
• A handful of examples is evidence, not a proof.
• State where each fact comes from: definition, hypothesis, or algebra.

How to practice until it clicks

Fluency comes from repetition on small claims, not from reading finished proofs. Start with statements whose only tools are the definitions of even and odd: the sum of two odds is even, an even times any integer is even, the square of an odd number is odd. Each one uses the exact skeleton above, so you drill the format while the content stays familiar.

When a proof feels stuck, return to the checklist: have I assumed the hypothesis, have I expanded every definition into symbols, and am I reasoning toward the conclusion rather than from it? After direct proofs feel natural, the next method to learn is mathematical induction, which handles claims about every natural number at once.