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.
16,111 questions
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How would one go about writing the proof to this limit convergence/complex sequence question? [closed]
Sorry for the super broad title. I’m not exactly sure where to start with this question for part (a).
(a) Suppose $\{s_n\}$ converges, and $\lim_{n \to \infty} s_n = s$. Prove that $\{ \operatorname{...
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Why does it make sense to approach the Basel problem this way?
I am putting together a presentation for my proofs class where I prove
$$
\sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6}.
$$
This proof from Proofs from the Book utilizes an evaluation of ...
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Connected space $X$ minus a component $C$ of the complement of a connected subset $A$, why is $X\setminus C$ still connected?
Let $X$ be a connected topological space and $A\subset X$ be a connected subspace of $X$. Let $C$ be a connected component of $X\setminus A$.
Is it true that $X\setminus C$ is connected?
This ...
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7
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Cleanest way to prove $\frac{\sin x}{1-\cos x} + \frac{1-\cos x}{\sin x} = \frac{2}{\sin x}$
I'm trying to prove the identity
$$\frac{\sin x}{1-\cos x} + \frac{1-\cos x}{\sin x} = \frac{2}{\sin x}$$
for all $x$ where both sides are defined.
My attempt. Combining the fractions on the left:
$$\...
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CLRS exercise: Upper bounding number of right rotations to transform a binary tree
This is exercise $13.2\text-5$ from Introduction to Algorithms.
If a binary tree with $n$ nodes $T_1$ can be turned into $T_2$ by performing a series of right rotations, then we say that $T_1$ can be ...
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$q:X\to Y$ be a quotient map with connected fibers. If $Z\subset Y$ connected, then so is $q^{-1}(Z)$. [duplicate]
I am trying to prove the following:
$If q:X\to Y$ be a quotient map with $q^{-1}(y)$ connected, for all $y\in Y$, and if $Z\subset Y$ is connected, then $q^{-1}(Z)\subset X$ is connected.
I have ...
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What axioms for addition are needed to prove commutativity when the naturals start at $1$ (no zero)?
I'm trying to prove that addition is commutative, but I'm working in a Peano-style setup where the naturals start at $1$ instead of $0$. Courant and John take $1$ as the first natural number in ...
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Examples of "Contradictory Subset" proofs
I noticed that certain proofs for Cantor's theorem and $\not\exists S, P(S)\subset S$ are incredibly similar, and I made a general outline for proofs of this form, that I've been giving many names (...
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Prove that $\frac{1+2\cos(\frac{\pi}{7})\cos(\frac{3\pi}{7})}{\sqrt{1+4\cos^2(\frac{\pi}{7})+4\cos(\frac{\pi}{7})\cos(\frac{3\pi}{7})}}=\cos(2\pi/7)$
Prove that $$\frac{1+2\cos\left(\frac{\pi}{7}\right)\cos\left(\frac{3\pi}{7}\right)}{\sqrt{1+4\cos^2\left(\frac{\pi}{7}\right)+4\cos\left(\frac{\pi}{7}\right)\cos\left(\frac{3\pi}{7}\right)}}=\cos\...
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To find $\left(\frac{3}{p}\right)$ [closed]
To find ($\frac{3}{p}$), $p>3$
Book says to use similar counting argument that is used in proof of $\left(\frac{2}{p}\right)$
Consider the integers
$$
1,2,3,\dots,\frac{p-1}{2}.
$$
Multiplying by $...
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Cutting a lidless cardboard unit-cubical box so that it folds flat, and then into three pieces that fit together to make a square
1996 Melbourne Uni Intermediate Schools Competition Q5
Question
You are given a cardboard cubical box without a lid. (Its sides and
bottom are squares of unit area.) Making as many straight line cuts
...
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Each onto function between compact topological Hausdorff spaces is irreducible under the restriction to some subset of its domain
I attempted solving question 17P(1) in Willard's book; consider some $X, Y$ compact Hausdorff spaces, and a continues surjective $f \colon X \to Y$. Then prove the existence of a compact $X_0 \subset ...
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How should I use verbs to introduce a new variable
This question is a part of a larger issue: how to write mathematical proofs that are both concise and accessible. Of course, this involves a trade-off, and we may never arrive at a universal solution.
...
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Is this proof that AI did the more standard text-book way of writing proofs, as opposed to the way that I did it? [closed]
I'm trying to self-learn how to write proofs in a more standard textbook way, though I know there is no one way to write a proof. I was using an abbreviated form of the rules of predicate calculus to ...
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A new proof of the irrationality of $\sqrt{2}$?
As far as I can tell, this is a proof that the set $\{a+b\sqrt{2}:a,b\in\mathbb{Z}\}$ is dense in $\mathbb{R}$ without using the fact that $\sqrt 2$ is irrational. But for every rational $r$, the set $...
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