(related to Part: Methods of Mathematical Proving)

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

- Set the context of the proof.
- Assert the hypothesis.
- List implications.
- State the conclusion.

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

- Set the context of the proof.
- Provide the hypothesis which will lead to a contrapositive of an original conditional statement.
- 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)."
- As a conclusion, restate the main implication.

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

- 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.
- You have now two possibilities to continue the proof:
- Possibility 1: Direct Proof:
- Assert the hypothesis, i.e. take $y$ and assume that $P(y)$ is true.
- List implications, showing that then $Q(y)$ must also be true.
- State the conclusion, e.g. "Therefore, it follows that $P\Rightarrow Q.$"

- Possibility 2: Proof by contradiction:
- Assert the hypothesis, i.e. take $y$ and assume that both: $P(y)$ is true and $Q(y)$ is false.
- List implications, leading to that contradiction.
- State the conclusion, e.g. "This is a contradiction to the hypothesis. Thus, it follows that $P\Rightarrow Q$".

- Possibility 1: Direct Proof:

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

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

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

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

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

- 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 $L.$
- 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 $0 < |x-a| < \delta.$

- List implications, i.e. show that $|f(x)-L| < \epsilon.$
- State the conclusion, e.g. "It follows by the definition of function limit that $\lim_{x\to a}f(x)=L$".

- 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 $L$.
- 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 $x > N$

- List implications, i.e. show that $|f(x)-L| < \epsilon$.
- State the conclusion, e.g. "It follows by the definition of function limit that $\lim_{x\to a}f(x)=L$."

- 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 $a.$
- 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 $2\epsilon.$

- 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.

- 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.

- 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 $a$.
- 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).

- 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 $|f(x)-L| < \epsilon=1$".

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016