# 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.

10,094 questions
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### How to find the numerical value of $\tan20\tan40\tan60\tan80$? [duplicate]

$$\tan20\tan40\tan60\tan80$$ I tried to bring $\tan20$, $\tan40$ in $\tan(a+b)$ form but couldn't get the answer. Using calculator I got it has $3$. Can anybody please tell me how to solve this?
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### Find the equivalent of a sequence [duplicate]

let $u_n$ such as $u_0 > 0$ and $u_{n+1}=u_n+1/u_n$ Show that $u_n \sim \sqrt{2n}$
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### Rigorous proof of a linear algebra theorem

I do seek a formal proof for the following statement. Let $V$ be a vector space such that $dimV=n$, let $S⊂V$ and let $v_1,...,v_r$ be a basis for $S$, then $S^⊥$ has $n-r$ linearly independepent ...
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### How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence

How can I prove that $\left\{\frac{1}{n^{2}}\right\}$ is a Cauchy sequence? A sequence of real numbers $\left\{x_{n}\right\}$ is said to be Cauchy, if for every $\varepsilon>0$, there exists a ...
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### Example for Hilbert quote

Hilbert famously said The art of doing mathematics consists in finding that special case which contains all the germs of generality. Can you give an example of a situation in mathematics where ...
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### Help: Hamming Code - Number of Parity Bits [PROOF]

I need to prove the following for the 1-bit correction hamming code by using equivalent transformations. $2^m (m+r+1) \leq 2^{m+r}$ => $r \geq 1 + \lfloor log2(m)\rfloor$ I have an Alphabet of size ...
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### Prove that all finite sets are countable [duplicate]

How can I prove that all finite sets are countable? A set $S$ is countable if there exists an injective mapping $f:\mathbb{N}\rightarrow S$. So, shall I prove the above by contradiction? I assume ...
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### Riemann integrals property proof [duplicate]

Prove that every function $f:[a,b]\rightarrow\mathbb{R}$ which is bounded and continuous everywhere but not in the finite number of points $x_1,...,x_n\in(a,b)$ has Riemann integral.
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### Prove that all subsets of countable sets are countable

This is basically a problem of my assignment, it says... A set is said to be countable if it is either finite or there is an enumeration(list) of the set. Then prove that All subsets of countable ...
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### The smoothing property of Bernstein polynomial

The smoothing property of Bernstein Polynomial, proved by Kelisky and Rivilin in 1967, $$\lim_{k\rightarrow\infty}B\;^{(k)}\left(f;x\right)=\left(1-x\right)f\left(0\right)+xf\left(1\right)$$ can be ...
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### Internal direct sum of kernel of surjective homomorphism and cyclic subgroup

I'm studying for a qualifying exam in algebra, and my abstract algebra skills are quite rusty. I'm attempting to solve the following problem: Suppose that $\Phi:G\rightarrow\mathbb{Z}$ is a ...
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### Finding the supremum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$ What is the Supremum of the above set? I consider the function $f(x)= x^{\frac{1}{x}}$, and show that $f(x)$ is maximum when $x=e$. ...
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### Going from local to global definition in proving sets are manifolds

I'm new to manifolds and in exercise 4.2.1 from J.J. Duistermaat's Multidimensional Real Analysis I, I have to prove that $V = \{x\in\mathbb{R}^2 : x_2=x_1^3\}$ is a $C^{\infty}$ manifold. The ...
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### Proving divisibility of integers [closed]

Given integers $x$ and $y$ and a prime number $k>3$. It turned out that $x + y$ and $x^2 + y^2$ are simultaneously divisible by $k$. Prove that $x^2 + y^2$ is divisible by $k^2$?
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### Subgroup of coprime order with automorphism group is contained in center of group

I'm studying for a qualifying exam in algebra and I've come across the following problem: Let $G$ be a finite group with a subgroup $N$. Let $Aut(G)$ be the group of automorphisms of $G$. Prove ...
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### Proof by contradiction on if and only if statements

Suppose I want to prove a general statement like 'A is true if and only if B is true' If I assumed B is untrue and showed that subsequently A is untrue, which direction am I actually proving? I guess ...
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### What's it called when, in a proof, we define a new variable/function in terms of two existing ones, in order to make it easier to write or follow?

Say, for instance, we're trying to prove a theorem involving two defined functions $f$ and $g$, and $x \in \mathbb{R}$. Within this proof, we have to deal with the sum $f(x) + g(x)$ multiple times. ...
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### Proving goal having the form $P \lor Q$, is it redundant to separate into two cases?

In Velleman's How to Prove It, the strategy given for proving goal of the form $P \lor Q$ goes like this: If $P$ is true, then clearly $P \lor Q$ is true. Now suppose $P$ is false. ...
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### Show set equality.

I know that to show set equality you must show that the two sets are subsets of each other. I'm having trouble showing that S\T is a subset of (S U T) given the assumption that T is a subset of S. I ...
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### what's the best way to prove the equivalences of such formulas?

I want to prove the following: $$2^n+2^{n-1}+...+2^1 + 1 = 2^{n+1}-1$$ The only Method that I know of is proof-by-induction but is this the best way to prove the equivalences of such formulas?
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### Proof that any integer $z>1$ can be written as $2x+y$, where $x>y$

Imagine a multiple choice questionnaire with 3 choices $a, b,$ and $c$. At the end the sums of each choice are tallied. It seems it's always possible to have a tie for first, as long as the total ...
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### Prove these differently written de morgan laws

We define $$\overline{\bigcup_{p\in P} Sp} =\bigcap_{p\in P} \overline{Sp}$$ and $$\overline{\bigcap_{p\in P} Sp} =\bigcup_{p\in P} \overline{Sp}$$ Which are just another way to write de morgans laws....
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### Showing that two numbers are the same percent different from their average.

More specifically, consider two real numbers $a,b>0$, and their average $r=\frac{a+b}{2}$. It is the case that $a=r*x$ and $b=r*y$ where $\vert 1-x\vert =\vert 1-y\vert$. For example, let $a=5$ ...
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### a < b if and only if a++ ≤ b.

I have to prove that a < b if and only if a++ ≤ b. I am using the book analysis 1 by Terence Tao, which unfortunately has no section for solutions of exercises. both a and b are natural numbers, ...
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### Proving $2\cosh 2x+ \sinh x = 5$ [closed]

I have been sitting on this question for quite some time and I haven't been able to prove this identity. Please anybody who can help me here. I am new to hyperbolics. $$2\cosh 2x+ \sinh x = 5$$ I ...
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### Prove infinite markov reward process converges

The following question is obtained from Stanford CS234 Lecture 2 notes, Excercise 3.7 Let $r_i$ denote the reward obtained from transition $s_i\rightarrow s_{i+1}$. Furthermore, the return $G_t$ of a ...
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### Prove that $0< \frac{1}{2^{m}} <y$

If $y$ be a positive real number, show that there exists a natural number $m$ such that $0< \frac{1}{2^{m}} <y$ I think I have to use Archimedean property to prove it. The Archimedean property ...
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### Formulating ordinary problems mathematically in order to solve them

I've been thinking about an ordinary problem for which there doesn't seem to exist a solution given its constraints. I was wondering how would one go about formulating the problem mathematically such ...
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### Two Proofs for Open Sets and Metric Subspaces

I have two proofs for the following theorem: Let $(S, d)$ be a metric subspace of $(M, d)$, and let $X$ be a subset of $S$. Then $X$ is open in $S$ if and only if $X = A \cap S$ for some set $A$ ...
How can I prove the inequality $\left|x\right|+\left|y\right|+\left|z\right|\le\left|x+y-z\right|+\left|y+z-x\right|+\left|z+x-y\right|$ for all $x, y, z$ being real number. Can I prove this by ...
### Find all pairs $(x, y)$ with $x, y$ real, satisfying the equations: $\sin\frac{(x+y)}2=0$ & $|x| + |y| = 1$
Find all pairs $(x, y)$ with $x, y$ real, satisfying the equations: $\sin\frac{(x+y)}2=0$ & $|x| + |y| = 1$ Working:$\frac{x+y}2=0$ or, $x=-y$ I plotted this. Plotting $|x| + |y| = 1$, I got ...