Proofs

Proof Templates:

the fancy english/structure you have to write to get the full points

In all cases it is good to write the info before starting the proof.

italic text is supposed to be filled in or replaced in the actual proof.

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Counterexample:

The statement is false for n = value goes here

(show why it is false for n = value)

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Proof by cases:

Make sure that the cases put together form the entire set you are trying to prove the proposition for.

info: $\text{For every}n\in \mathbb{R},$ S(n): (...)

Proof: By cases we distinguish two cases

Case one: (...)

Case two: (...)

In both cases we proved S(x), since for each $x\in \mathbb{R}$ we have either case one or case two we proved that S(x) holds for all $x\in \mathbb{R}$$◻$

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Direct proof:

info: $\text{For every}n\in N,$ S(n): (...)

Proof: take an arbitrary integer x such that x fufills conditions$◻$

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Proof by contradiction:

info: P -> Q

Proof: Assume, for the sake of contradiction P is true but Q is false.

(...)

Since we have a contradiction, it must be that Q is true.

$◻$

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Proof by Induction:

info: $\text{For every}n\in N,$ S(n): (...)

PROOF: By mathematical induction.

Basis Step: S(1) asserts that (...)

which is true because (...)

Inductive Step:

Assume for an arbitrary $k\in N$, S(k) is true, i.e.: (...) Induction Hypothesis (substitute k for n)

We will now show that S(k+1) is also true, i.e.: (...) state what must be proved (substitute k+1 for n)

Proof of inductive step: (...)

Conclusion: We thus have that S(1) is true and $\mathrm{\forall }k\in N,S(k)⟹S(k+1)$, so by the principle of mathematical induction, it follows that S(n) is true for all natural numbers n. $◻$

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Sources

• Harry Aarts, Ed Brinksma, Jan Willem Polderman, Gerhard Post, Marc Uetz, Marjan van der Velde (2018) Introduction to Mathematics
• Micro-lectures wk2
• villanova-induction-template-pdf