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.

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Prove that for any real numbers $x$ and $y$ if $x \neq 0$, then if $y=\frac{3x^2+2y}{x^2+2}$ then $y=3$.

I've been reading Velleman's How to Prove it and I'm having a good time with the book up to now. However, I've been really stuck in the exercice 10 (Ch 3, Sec 3.2): Prove that for any real numbers $x$ and $y$ if $x \neq 0$, then if…
Bruno Bentzen
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Property of a $\sigma$-additive function

Let $X$ be a set and ${\cal A}\subset 2^X$ a ring. If $\mu:{\cal A}\to [0,+\infty]$ is a $\sigma$-additive function and $(A_i)_{i\in\Bbb N}$ is a family in $\cal A$ such that $\bigcup_{i=1}^\infty A_i$ belongs to $\cal A$, then does the…
User
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Proving injective (1-1) using contrapositive

Given function $f:\mathbb Z \to \mathbb Z$ defined by $f(n) = n - 6$ $\mathbb Z$ in this case is the set of integers. Suppose for $x_1$, $x_2 \in \mathbb Z$, we have $f(x_1) = f(x_2)$. This means that $x_1 - 6 = x_2 - 6$ Hence $x_1 = x_2$. By the…
jn025
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Prove $x^2=t$ for any $t>0$

Prove for any positive number $t$, there is a solution for $x^2=t$. So we want to show that $x^2=t$ for $t\geq0$. We can break this into two cases: Case 1: Assume $t=0$, then we have $x^2=0$ $\Rightarrow$ $x=0$, and we are done. Case 2: Assume…
Michelle
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Prove that if g is injective, f is injective

$f \colon A \to \mathbb R$ be a function (where $A$ is some set) and define the function $g \colon A \to \mathbb R$ as $g(x) = 3 (f(x))^2 + 1.$ Prove if $g$ is injective then $f$ is injective How do I prove the composite of the function is…
user3201346
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Decomposition of a function in positive and negative parts and its integrability

Is it true to say that $\int_\mathbb{R}|f(x)|dx<\infty\Rightarrow\int_\mathbb{R}f(x)=0$?
SAMEER
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When writing a proof, why do we want to assume a different but equivalent condition given in the proposition?

In the proof for the inductive step, we start by assuming $k \ge 10$. But along the way, the author mentions $k \ge 1$ and $k \ge 7$ to justify the inequality. Why do we bother to do this instead of just sticking with $k \ge 10$?
mauna
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Proof regarding division with a remainder

Let $a\in\mathbb{Z},n\in\mathbb{N}$. If $a$ has a remainder $r$ when divided by $n$, then $a\equiv r\pmod n$ I've done some of these questions before with modulus and division, but I'm unsure of how to approach it with the addition of a remainder
user158123
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Modulus related proof help

I need to prove this via either direct proof, or contrapositive. Unsure of the best way to approach this. if $a \equiv b\mod n$ and $c \equiv d\mod n$, then $ac \equiv bd\mod n$ So far I have: Suppose $a \equiv b\mod n$ and $c \equiv d\mod n$, then…
Wilson
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Help with this proof

If the sets $A$ and $B$ are bounded above and $A\subseteq B$ and $A$ and $B$ both have supremums, then $sup(A)\le sup(B)$ Came across it in my textbook and was wondering how to prove it. It looks pretty simple. Thanks!
user135340
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Proof that in any base, the sum of two numbers of fixed precision can't have carry more than 1

The best proof that I came with is, given any base $b$, let $c$ be the greatest number can be written whit $n$ digits. Then the number will be: $$c=b^0(b-1)+b^1(b-1)+\cdots +b^n(b-1)$$ Summing this number twice, I'll get the maximum possible carry,…
FranckN
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What is this problem stating? And how to prove this?

$$\exists! x : A(x) \Rightarrow \exists x : A(x)$$ Assuming that $A(x)$ is an open sentence. I'm new to abstract mathematics and proofs, so I came here to ask for some simplification. Thanks
user2628815
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Proof for a function $m:2^S\to R$

Let $S$ be a finite, non-empty set and $m:2^S\to R$ a function with the following properties $M1$: $\forall A\in2^S, m(A)\ge0$ $M2$: $\forall A, B\in 2^S, A\cap B=\varnothing\Longrightarrow m(A\cup B)=m(A)+m(B)$ $(a)$ Prove that $m(\varnothing) =…
user136038
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Validity of using "and so on" in a proof for finite number of iterations

I want to prove that for a metric space $(\frak{M},\rho)$ if $M\subset \frak M$ satisfies $$M\subset \mathop{\bigcup}_{k=1}^n B[x_k,r_k]\;\;(n\in\mathbb N),$$ then $M$ is a bounded set. Here's my attempt at proving this: Take balls $B_1=B[x_1,r_1],…
Ruslan
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Help with this proof (Index Sifting)

Let $(x_j)^\infty_{j=1}$ be a sequence in $\mathbb{Z}$ and let $a, b, r \in \mathbb{Z}$ such that $a\le b$. Then $\sum\limits_{j=a}^b x_j = \sum\limits_{j=a+r}^{b+r} x_{j-r}$ I assume that I would use induction to solve this, maybe on r?
user3182252
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