Writing Proofs

Why “writing” a proof is its own skill

Knowing proof techniques (direct proof, contradiction, induction, etc.) is not the same as being able to write a clear, convincing proof that a reader can follow. Writing proofs involves:

• Choosing what to say and what to omit.
• Ordering the ideas so the logic is easy to follow.
• Expressing the argument in precise language and notation.

Here we focus on how to turn a mathematical argument into well-structured written text.

General structure of a proof

Most proofs, regardless of technique, follow the same overall pattern:

1. Set up the statement you are proving.
2. State your assumptions clearly (what you are allowed to use).
3. Develop the logical argument step by step.
4. Finish with a clear conclusion that matches exactly what you were asked to prove.

It is often helpful to imagine the following “template”:

• Restate the goal (sometimes briefly, in your own words).
• Assume the hypotheses (the “given” conditions).
• Reason clearly from the hypotheses using definitions, known theorems, and logical steps.
• Arrive at the conclusion (the thing to show), and explicitly say that you have reached it.

For example, if you are proving a statement of the form

$$\text{If } P \text{ then } Q,$$

a typical proof structure is:

1. “Assume $P$.” (Set up your starting point.)
2. Reason step by step using definitions and previous results.
3. “Therefore $Q$.” (Make it explicit when you reach the goal.)

What to include and what to omit

Writing a proof is partly about judging the right level of detail.

• Include:
• Definitions you actually use.
• Key justifications for non-obvious steps.
• Clear connections between one line and the next.
• Omit:
• Trivial arithmetic steps (like $2+3=5$) unless they are the entire point.
• Long digressions that are not part of the proof.
• New, unproven claims that are not justified.

A useful rule of thumb:
Write your proof so that a careful reader at your level could follow it without guessing your intentions, but without being bored by unnecessary micro-steps.

Organizing your thoughts before writing

Before you start writing full sentences, it often helps to:

1. Sketch the logical idea in rough form.
• What is given?
• What must be shown?
• Which proof technique will you use?
• Which definitions or theorems are likely relevant?
2. Work out the “math” scratch work first.
• Do computations.
• Try examples.
• Look for patterns.
• Find a chain of reasoning that gets you from assumptions to conclusion.
3. Then translate this into a clean, linear argument.
• Remove false starts and dead ends.
• Keep only the successful path, written clearly.

The final written proof should look as if you thought of it in a straight line, even though, in reality, you may have tried many different approaches.

Typical formats for common types of statements

Different logical forms suggest different writing patterns. Here we focus on how you write them, not on how to prove them in detail.

Proving “if … then …” statements

A statement of the form “If $P$, then $Q$” (often written $P \Rightarrow Q$) is usually written like this:

“Proof. Suppose $P$ is true.
[Reasoning using $P$, definitions, and known results.]
Therefore, $Q$ is true. $\square$”

Key writing points:

• Use words like “Assume”, “Suppose”, or “Let” to introduce the hypothesis.
• Make it clear when you are switching from assumptions to deductions.
• End with an explicit statement: “Hence $Q$ holds,” or “This proves the implication.”

Proving “for all” statements

Many theorems start with “For all $x$ in some set, …” or “For every integer $n$, …”. The typical structure is:

“Theorem. For every integer $n$, [some property $P(n)$ holds].

Proof. Let $n$ be an arbitrary integer.
[Show that $P(n)$ is true, using only that $n$ is an arbitrary element of the set.]
Since $n$ was arbitrary, the property holds for all integers $n$. $\square$”

Important writing ideas:

• The word “arbitrary” matters: you are not picking a special $n$, but a generic one.
• Do not use any special property of $n$ that is not true for all integers (unless it is part of the hypotheses).
• At the end, explicitly connect the argument for the arbitrary element to the universal claim “for all”.

Proving “there exists” statements

For a statement “There exists an $x$ such that $P(x)$ holds”, the usual pattern is:

“Proof. We must show that there is at least one $x$ such that $P(x)$ is true.
Consider $x = [some explicit choice]$.
[Show that $P(x)$ holds for this choice.]
Therefore, such an $x$ exists. $\square$”

Key writing points:

• Usually you exhibit a specific example: “Take $x = 3$” or “Define $f(x) = x^2$”.
• Make it explicit that your example satisfies the required property.
• If the existence proof is non-constructive (you know something exists but you do not describe it explicitly), clearly explain why existence follows from a theorem, not just that you “believe” it.

Proving “if and only if” statements

A statement “$P$ if and only if $Q$” (written $P \Leftrightarrow Q$) means:

• “If $P$ then $Q$” and
• “If $Q$ then $P$.”

The structure is usually:

“Theorem. $P$ if and only if $Q$.

Proof.
($\Rightarrow$) First, suppose $P$ holds.
[Show that $Q$ follows.]

($\Leftarrow$) Now suppose $Q$ holds.
[Show that $P$ follows.]

Therefore, $P$ holds if and only if $Q$ holds. $\square$”

Writing advice:

• Clearly separate the two directions with labels like “($\Rightarrow$)” and “($\Leftarrow$)” or with phrases like “For the converse”.
• Each direction should read like a normal “if … then …” proof.
• At the end, explicitly state that you have proved both directions.

Proving a statement by cases

Sometimes a statement is naturally divided into cases (for example, even vs. odd integers, positive vs. non-positive numbers). A written case proof might look like:

“Proof. We consider two cases.

Case 1: $n$ is even.
[Argument showing the statement holds when $n$ is even.]

Case 2: $n$ is odd.
[Argument showing the statement holds when $n$ is odd.]

In both cases, the statement is true. Therefore, the statement holds for all integers $n$. $\square$”

Writing advice:

• Number or label your cases clearly.
• Make sure the cases cover all possibilities (and mention that they do).
• In each case, be explicit about which assumptions you are using (for example, “since $n$ is even, we can write $n = 2k$ for some integer $k$”).

Using definitions effectively in writing

Definitions are the building blocks of proofs. Good proof writing makes the use of definitions visible and explicit. Often this involves:

• Unpacking a definition when needed:
• Example: “By definition, an even integer is one that can be written as $2k$ for some integer $k$.”
• Referring back to the definition to justify a step:
• Example: “Thus $n = 2k$, so by definition $n$ is even.”

It is not necessary to rewrite the full formal definition every time, but at key steps you should:

• Clearly indicate when you are using a definition.
• Use consistent notation that matches the definition.

A common pattern is:

• Start with the hypothesis: “Let $n$ be an even integer.”
• Use the definition: “Then, by definition, $n = 2k$ for some integer $k$.”
• Continue the proof using that representation.

Justifying steps: linking each line to a reason

A clear proof helps the reader see why each step is valid. You can do this by:

• Using connecting phrases:
• “Since …”
• “Because …”
• “Therefore …”
• “Hence …”
• “It follows that …”
• Occasionally naming the reason:
• “By the definition of …”
• “By the triangle inequality, …”
• “Using the fact that …” (referring to a previously proved theorem or lemma).

For example:

“Since $n$ is even, we can write $n = 2k$ for some integer $k$. Therefore $n^2 = (2k)^2 = 4k^2$, which is divisible by 4. Hence $n^2$ is even.”

Each “since”, “therefore”, and “hence” tells the reader that the statement is not random: it is a logical consequence of what came before.

Notation and sentences

Mathematical writing uses a mixture of words and symbols. Good proof writing:

• Uses symbols where they clarify, not where they confuse.
• Integrates symbols into sentences with proper grammar.
• Avoids long strings of symbols without explanation.

Compare:

• Poor:
  “$n$ even $\Rightarrow \exists k \in \mathbb{Z}, n=2k \Rightarrow n^2=4k^2$ even.”
• Better:
  “Suppose $n$ is even. Then there exists an integer $k$ such that $n = 2k$. Hence
  $$n^2 = (2k)^2 = 4k^2,$$
  which is an even integer.”

Tips:

• Start sentences with words, not symbols, when possible.
• Treat equations like parts of a sentence:
• “We have $n^2 = 4k^2$,” not just “$n^2=4k^2$”.
• Use punctuation around formulas as if they were ordinary words. A displayed equation often ends with a comma or a period, depending on the sentence.

Common patterns in proof introductions

The first sentence of a proof often signals the method and the assumptions. Some useful opening phrases:

• Direct proofs:
• “Let $x$ be a real number with [property].”
• “Suppose $n$ is an integer such that [condition].”
• Proofs by contrapositive:
• “We will prove the contrapositive: if not $Q$, then not $P$.”
• “Instead of proving $P \Rightarrow Q$ directly, we show that $\lnot Q \Rightarrow \lnot P$.”
• Proofs by contradiction:
• “Suppose, for the sake of contradiction, that the statement is false.”
• “Assume, contrary to our goal, that [negation of the conclusion].”
• Proofs by induction:
• “We proceed by induction on $n$.”
• “First, we verify the base case. Then we prove the inductive step.”

Even though the detailed techniques are treated elsewhere, these phrases help the reader immediately understand what kind of argument is coming.

Finishing a proof clearly

A good proof does not just stop when the writer sees the conclusion; it announces that the argument is complete.

Typical closing phrases:

• “Thus $Q$, as required.”
• “Therefore, the statement holds for all $n$.”
• “This completes the proof.”
• Followed by a proof-ending symbol, like $\square$ or “QED”.

This makes it easy for the reader to see where the proof ends, and confirms that you have actually proved the original claim, not just something similar.

Common mistakes in writing proofs (and how to fix them)

Here are some frequent issues in beginner proofs, with writing-focused remedies.

Using examples instead of proof

Mistake:
“To show that every even number is divisible by 2,” a student writes: “$4$ is divisible by 2, $6$ is divisible by 2, and $8$ is divisible by 2, so it must be true.”

Fix in writing:

• A proof needs a general argument, not a list of examples.
• Write something like: “Let $n$ be an even integer. Then $n = 2k$ for some integer $k$, so $n$ is divisible by 2.”

Assuming what you want to prove

Mistake:
In trying to prove “Every multiple of 4 is even,” a student writes: “Assume every multiple of 4 is even. Let $n$ be a multiple of 4, then $n$ is even. So the statement is true.”

Fix:

• Do not assume the whole statement; start only from the hypothesis about $n$.
• Write: “Let $n$ be a multiple of 4. Then $n = 4k$ for some integer $k$, so $n = 2(2k)$ is even.”

Vague phrases

Mistake:
“Clearly, it’s obvious that $n^2$ is even,” or “It is trivial.”

Fix:

• Avoid words like “clearly”, “obviously”, or “trivially” unless the step is truly immediate (like recognizing that $n^2 \ge 0$ for real $n$).
• If a step might not be clear to a reader at your level, show the reasoning instead of naming it obvious.

Broken logic and missing links

Mistake:
Writing down correct statements, but not connecting them:

“Let $n$ be even. Then $n=2k$. Also, $4$ divides $8$. Thus $n$ is even.”

Fix:

• Every step should be a consequence of the previous ones and lead toward the conclusion.
• Remove irrelevant statements.
• Use connecting words like “therefore” or “hence” after genuinely derived statements, not randomly.

Revising and improving a proof

After writing a draft proof, it is worth revising it at least once. A simple checklist:

1. Does the proof match the statement?
• Compare the final line of your proof to the original claim.
2. Are all assumptions stated?
• Did you clearly say “Let $n$ be an integer …”, “Assume $x > 0$”, etc.?
3. Are all non-trivial steps justified?
• Can you say, for each line, why it is true based on previous lines, definitions, or known results?
4. Is the structure easy to follow?
• Are cases clearly separated?
• Are directions in an “if and only if” proof labeled?
5. Is the language clear and not overly symbolic?
• Could you replace some long symbolic chains with a sentence?

As you gain experience, you will spend less time on these checks, but at the beginning they help establish good habits.

Putting it together: from idea to polished proof

To summarize the process of writing proofs:

1. Understand the statement.
• Identify its form: “if … then …,” “for all,” “there exists,” “if and only if,” etc.
2. Decide on a proof strategy.
• Direct, contrapositive, contradiction, induction, by cases, etc. (covered in the chapter on proof techniques).
3. Do rough work.
• Try out ideas and find a logical chain from assumptions to conclusion.
4. Write a clear, linear argument.
• Begin with assumptions using precise language.
• Use definitions and theorems with explicit justifications.
• Keep the logic forward-moving and connected.
5. Conclude explicitly.
• State that you have reached the required conclusion and end the proof.

With practice, your proofs will become not only correct, but also readable, well-structured, and convincing to others.