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Explanation: Good Practices for Writing Mathematical Proofs

(related to Part: Methods of Mathematical Proving)

Many proofs can be written by following some standard guidelines or good practices to write a proof.

Basic Template for Proofs

1. Set the context of the proof.
2. Assert the hypothesis.
3. List implications.
4. State the conclusion.

Template for Proofs by Induction

1. Set the context of the proof.
2. Base Case: Prove that a statement $S$ is true for the base case $S(b).$
3. Hypothesis: Assume that $S(n)$ is true for some fixed natural number $n:=k\ge b.$
4. Induction Step: Using the fact that $S(n)$ is true, prove that $S(n+1)$ is true.
5. State the conclusion by induction, i.e. conclude that $S(n)$ is true for all natural numbers $n\ge b$.

Proof by Contraposition

1. Set the context of the proof.
2. Provide the hypothesis which will lead to a contrapositive of an original conditional statement.
3. Provide the implication, e.g. "Since (refer here to the original conditional statement), it follows from the hypothesis by contraposition that (provide here the implication)."
4. As a conclusion, restate the main implication.

Proof by Contradiction:

1. Set the context of the proof.
2. Assert the hypothesis that will lead to a contradiction.
3. List implications, leading to that contradiction
4. State the conclusion, e.g. "This is a contradiction to the hypothesis."

Proof of an implication

1. Set the context of the proof, i.e. state that you want to prove the implication $P\Rightarrow Q$, in which you replace $P,Q$ by two statements, for which you want to show the implication.
2. You have now two possibilities to continue the proof:
   • Possibility 1: Direct Proof:
     1. Assert the hypothesis, i.e. take $y$ and assume that $P(y)$ is true.
     2. List implications, showing that then $Q(y)$ must also be true.
     3. State the conclusion, e.g. "Therefore, it follows that $P\Rightarrow Q.$"
   • Possibility 2: Proof by contradiction:
     1. Assert the hypothesis, i.e. take $y$ and assume that both: $P(y)$ is true and $Q(y)$ is false.
     2. List implications, leading to that contradiction.
     3. State the conclusion, e.g. "This is a contradiction to the hypothesis. Thus, it follows that $P\Rightarrow Q$".

Proof of Equality of Sets

1. State the assumption that e.g. $A$ and $B$ are sets.
2. Part 1: Provide a proof of $A\subseteq B$, e.g. using one of the other proof templates.
3. Part 2: Provide a proof of $B\subseteq A$, e.g. using one of the other proof templates.
4. State the conclusion, e.g. "It follows from the equality of sets that $A = B$".

Proof of Inclusion of two Sets $A\subseteq B$

1. Set the context of the proof, i.e. what is being assumed about the sets $A$ and $B$.
2. Assert the hypothesis, e.g. "let $x\in A$".
3. List implications, in particular, use the properties of the set $A$ and $B$ to show $x$ belongs to the set $B$.
4. State the conclusion, e.g. "It follows that $x\in B$. Thus, by the definition of subset, $A\subset B$.

Proof of a function $f$ being surjective

1. Set the context of the proof, i.e. introduce the function $f: A\to B$, its domain $A$ and its domain $B$.
2. Assert the hypothesis, i.e. select an arbitrary element of the codomain, e.g. "let $y\in B$".
3. List implications, i.e. exhibit a value $x\in A$ with $f(x)=y.$
4. State the conclusion, e.g. "It follows by the definition of surjective functions that $f$ is surjective".

Proof of a function $f$ being injective

1. Set the context of the proof, i.e. introduce the function $f: A\to B$, its domain $A$ and its domain $B$.
2. Assert the hypothesis, i.e. select two values $x_1,x_2\in A$ and assume that $f(x_1)=f(x_2)$.
3. List implications, i.e. show that $x_1=x_2.$
4. State the conclusion, e.g. "It follows by the definition of injective functions that $f$ is injective".

Proof of a function $f$ being bijective

1. Set the context of the proof, i.e. introduce the function $f: A\to B$, its domain $A$ and its domain $B$.
2. Prove that $f$ is injective (you might use the template listed above for injective functions).
3. Prove that $f$ is surjective (you might use the template listed above for surjective functions).
4. State the conclusion, e.g. "It follows by the definition of bijective functions that $f$ is bijective".

Proof that a number $L$ is the limit of a function $f(x)$, as $x$ tends to a value $a.$

1. Set the context of the proof, i.e. introduce the function $f: A\to B$, its domain $A$ and its domain $B$ and say what is known about the numbers $a$ and $L.$
2. Assert the hypothesis, i.e.:
   • Select an arbitrary positive number $\epsilon$, e.g. "Given $\epsilon > 0.$"
   • Propose an appropriate value for $\delta$, e.g. "Let $\delta:=\ldots$"
   • Select $x$ such that $0 < |x-a| < \delta.$
3. List implications, i.e. show that $|f(x)-L| < \epsilon.$
4. State the conclusion, e.g. "It follows by the definition of function limit that $\lim_{x\to a}f(x)=L$".

Proof that a number $L$ is the limit of a function $f(x)$, as $x$ tends to infinity $\infty.$

1. Set the context of the proof, i.e. introduce the function $f: A\to B$, its domain $A$ and its domain $B$ and say what is known about the number $L$.
2. Assert the hypothesis, i.e.:
   • Select an arbitrary positive number $\epsilon$, e.g. "Given $\epsilon > 0$."
   • Propose an appropriate value for a natural number $N$.
   • Select $x > N$
3. List implications, i.e. show that $|f(x)-L| < \epsilon$.
4. State the conclusion, e.g. "It follows by the definition of function limit that $\lim_{x\to a}f(x)=L$."

Proof that the limit of a function $f(x)$ does not exist, as $x$ tends to a value $a.$

1. Set the context of the proof, i.e. introduce the function $f: A\to B$, its domain $A$ and its domain $B$ and say what is known about the number $a.$
2. Assert the hypothesis, i.e.:
   • Assume that $L$ is the limit, i.e. $\lim_{x\to a}f(x)=L.$
   • Propose an appropriate value for a positive number $\epsilon$, e.g. "Let $\epsilon:=\ldots$"
   • Select an arbitrary value for $\delta$, e.g. "Given $\delta > 0.$"
   • Select appropriate values $x_1,x_2$, i.e. such that $0 < |x_1-a| < \delta,$ $0 < |x_2-a| < \delta,$ and that $|f(x_1)-f(x_2)|$ exceedes $2\epsilon.$
3. List implications, i.e.
   • Assume that $|f(x_1)-L| < \epsilon$ and that $|f(x_2)-L| < \epsilon.$
   • Using triangle inequality, show that $2\epsilon =\epsilon + \epsilon > |f(x_1)-L|+|f(x_2)-L|=|f(x_1)-L|+|L-f(x_2)|\ge |f(x_1)-L+L-f(x_2)|=|f(x_1)-f(x_2)|.$
   • State the contradiction, e.g. "This shows that $2\epsilon > |f(x_1)-f(x_2)|,$ which is a contradiction.
4. State the conclusion, e.g. "Thus, it cannot hold that both $|f(x_1)-L| < \epsilon$ and that $|f(x_2)-L| < \epsilon.$ Therefore, the limit $L$ does not exist.

Proof that the limit of a function $f(x)$ does not exist, if $f$ is unbounded as $x$ tends to a value $a.$

1. Set the context of the proof, i.e. introduce the function $f: A\to B$, its domain $A$ and its domain $B$ and say what is known about the number $a$.
2. Assert the hypothesis, i.e.:
   • Assume that $L$ is the limit, i.e. $\lim_{x\to a}f(x)=L$.
   • Propose "Let $\epsilon:=1$ and let $\delta > 0$ be given."
   • Select an appropriate number $x$ such that $0 < |x-a| < \delta$ (but suitable for implications to follow).
3. List implications, i.e.
   • Show that $f(x) > |L| + 1.$
   • Conclude that $|f(x)-L| > |L|+1-L \ge 1.$
   • State the contradiction, e.g. "This is a contradiction to $|f(x)-L| < \epsilon=1$".
   • State the contradiction, e.g. "This is a contradiction to $|f(x)-L| < \epsilon=1$".

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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016