Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

Questions with this tag are about the presentation of a mathematical proof. Questions might include:

  • Should I include [x-mathematical detail] at [y-part of this proof]?
  • Is the following a sufficient proof of [x-mathematical tidbit]?
  • I have written the following proof, could I somehow improve it, does it have good flow/can I improve readability?

But this tag is not for asking someone else to write a proof for you, or for how to answer some question. Questions such as: My professor asked me to prove the Pythagorean theorem and I don't know how to begin are not to have this tag.

This tag is intended for use along with other, more "mathematical" tags. A question about the writing of a proof in abstract algebra, for example, should have as well. This tag can be used along with the proof verification tag.

See here for a useful set of guidelines for writing a solution.

15776 questions
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How to formalize this phrase from an informal proof?

How do I formalize this phrase from an informal proof using some form of natural deduction: We claim that X. To obtain a contradiction, assume that Y. https://digitalcommons.kennesaw.edu/cgi/viewcontent.cgi?article=2161&context=facpubs (p.2 of…
Dan Christensen
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Application of mathematical induction to divisibility

Prove by induction that $6^n + 4$ is divisible by $5$. Using another method I found that the solution is $5\cdot6^k$ when i saw the solution using induction they made an assumption, where $6^k + 4 = 5m$, reaching, $6\cdot6^k + 4$. Their final…
Maxwell
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Is there a unified theorem index?

I'm currently reading a book on analysis and always refer to specific theorems (e.g. chapter 2, theorem 3) when I need to prove something in the exercises. Looking at a different analysis book, the same theorems have (of course) different…
itmuckel
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Cauchy sequence and a particular subsequence

Let $\{{f_n}\}$ a Cauchy sequence. Then by definition $$\forall k\in\mathbb{N},\quad\exists n(k)\in\mathbb{N}\quad|\quad\forall m,n>n(k)\quad \|f_m-f_n\|<\frac{1}{2^k}$$ I must prove that we can choose a increasing sequence of natural numbers…
Jack J.
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Let $z\in\mathbb{Z}$. Then $7\mid z$, if and only if $7\mid z^2$

I know this is a proof by cases, and it should be proved directly first, then conversely we find the converse, and then find the contrapositive of the converse and prove that. Would our cases be $7$ divides $z$, and $7$ does not divide $z$? I also…
gkc52
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Proof in Theorem List of L

Prove the theorem $\vdash A \rightarrow (\neg B \rightarrow \neg(A \rightarrow B))$ I proved a variety of other theorems in the Theorem List that my university professors created…
Erika
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How would you prove that subtraction can be deferred to follow addition?

Addition is commutative, which easily proves that (freely speaking) the order of summing a list of numbers does not matter; all orders of applying the summation are equal. How would you go about proving, from a notion of minimalist "first…
matanox
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Say I have a function $f(x) = e^x -2$ with $x \in [0, 1]$. How do I show that $f(0) < f(x) < f(1)$ $\forall x \in [0,1]$.

I know this is quite a basic question, but at this current time I can't think of a proof. Here's the question. Say I have a function $f(x) = e^x -2$ with $x \in [0, 1]$. How do I show that $f(0) < f(x) < f(1)$ $\forall x \in [0,1]$. Could someone…
user486957
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Let A, B and C be sets. Prove that $A \times (B-C) = (A \times B) - (A \times C)$

Because of distributive properties we know this is true, but I am stuck on showing the proof. I know: Let $x\in A\times (B-C)$. This means $x\in\mathbb{A}$ and $x\in B$ but $x\ not in C$. Assume $x\in A$ then from here im not sure where to go.
user774710
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Existence of a result on sequences

Let $\Omega\subseteq\mathbb{R}^n$ be an open set. If $x\in\overline{\Omega}$ Can I always find such a sequence $\{x_n\}\subseteq\Omega$ such that $x_n\to x$? I believe such a result exists, but I don't remember if the hypotheses and why it can be…
Jack J.
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If $x>0$ then prove that $\frac{1}{x}> 0$.

I want to go about doing it instead of like putting numbers in. So the professor did it like this: $\textbf{Proof:}$ Since $x>0$, let $x$ = $5$ Then $\frac{1}{5}$ > $0$ Proof End. But I don't know, I don't find it satisfying enough I suppose. Is…
Arkilo
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Conceptual question on proving uniqueness

I was always taught that if I want to prove that some element, say an additive inverse, is unique, that I suppose there are two inverses and establish that they are equal. What was left out I think, though, was the specific proof strategy. It seems…
user465188
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Finding the image of $f: (0, \infty) \to \mathbb{R}, f(x) = \frac{4x}{x+1}$

I need to find the image of the function $f: (0, \infty) \to \mathbb{R}, f(x) = \frac{4x}{x+1}$. What I did was make the function in terms if y to end up with: $x = \dfrac{y}{4-y}$ Now, I don't know how to go from here. I know that y can't equal…
bob657
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if $\nu(E) = \sum_{k=1}^{n} c_{k} \mu(E \bigcap E_{k}).$ Show that $\nu$ is a measure on $(X, \mathfrak{M})$

Let $(X, \mathfrak{M}, \mu)$ be a finite measure space, let $\{E_{k}\}_{k=1}^{n}$ a collection of measurable sets, and $\{ c_{k}\}_{k=1}^n$ a collection of real numbers. For $E \in \mathfrak{M}$ define $$\nu(E) = \sum_{k=1}^{n} c_{k} \mu(E \bigcap…
Emptymind
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