# 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.

15,729 questions
Filter by
Sorted by
Tagged with
23 views

### Real Number recursion

Let $T_{n}$ be a sequence of real numbers such that $T_{n+1}\ge T_{n}^{2}+\frac{1}{5}$ Prove that $\sqrt{T_{n+5}}\ge T_{n}$. I rearranged the first expression in terms of $T_{n}$, but seemingly to no ...
20 views

46 views

### How do I go about proving if a|b then $a^n|b^n \forall n \in \mathbb{N}$

I'm trying to prove that if a|b then $a^n|b^n, \forall n \in \mathbb{N}$. I know $a|b \rightarrow b = aq, q \in \mathbb{Z}$ by the definition of divisibility. I can then (I think) exponentiate both ...
1 vote
102 views

### How to argue that two outcomes are equally likely?

When it comes to the concept of "equally likely," students are not told what makes two outcomes equally likely, but are instead shown what it means via examples. "When flipping a coin, ...
8k views

### Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too [duplicate]

I'm trying to figure out how to prove that if $p$ and $p^2+2$ are prime numbers then $p^3+2$ is a prime number too. Can someone help me please?
3k views

### proving that the area of a 2016 sided polygon is an even integer

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90$°or $270$°. If the lengths of its sides are odd integers, prove ...
2k views

3k views

### Prove that $\sqrt{8}$ is irrational in different method

I tried to prove that $\sqrt{8}$ is irrational. I said let $\sqrt{8}$ be rational then $\sqrt{8}$ = $a/b$ where $a$ and $b$ are relatively prime. Then $2\sqrt{2}=a/b$ , and $\sqrt{2} =a/(2b)$. it is ...
4k views

### Prove If $a \equiv b\pmod{n}$ and $c\equiv d\pmod{n}$, then $a + c \equiv b + d \pmod{n}$ [closed]

Using the definition of congruence $\pmod{n}$, prove that if $a \equiv b\pmod{n}$ and $c \equiv d \pmod{n}$, then $a + c \equiv b + d \pmod{n}$. I've seen a couple of different examples of how to do ...
43 views

### Given two linear transformations T1 and T2, show that range T1 = range T2 if and only if there is an invertible operator S such that T1=T2S.

I am working through Axler's Linear Algebra Done Right section 3D problem 5 which states "Suppose $V$ is finite-dimensional and $T_1, T_2 \in \mathcal{L}(V,W)$. Prove that range $T_1=$ range $T_2$...
97 views

### Is $(\phi\implies\psi) \iff ((\phi\land\psi)\lor \neg\phi)$ provable in the following hilbert calculus without contraposition or reductio?

Is $(\phi\implies \psi) \iff ((\phi\land \psi)\lor \neg \phi)$ in the following hilbert calculus without contraposition or reductio ad absurdum? If so, how would I go about proving it, and if not, ...
32 views

### Is my proof using induction correct. $(\forall n \in \mathbb{N}, n\geq 5) 2^{n+3} < (n+1)!$

I'd appreciate if somebody could check whether my proof (mathematical induction) is correct. $$(\forall n \in \mathbb{N}, n\geq 5) 2^{n+3} < (n+1)!$$ (i) let $n=5$ $2^{5+3} = 2^{8} < (5+1)!=6!$ (...
1 vote
1k views

### Proving set statements

Let A = {x ∈ Z: x = 5a + x for some integer a} B = {y ∈ Z: y = 10b - 3 for some integer b} C = {z ∈ Z: z = 10c + 7 for some integer c} Prove or disprove the ...
762 views

### Proving $\sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.

Give an algebraic and a combinatorial proof for the following identity: $$\sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}.$$ For the combinatorial argument, use the analogy ...
1 vote
21 views

### $P (A_1)= P (A_1\cap A_2)+ P(A_1\cap A_2^c)$ (Proof verification)

Is my proof well writen and correctly stated? Let $(\Omega,\mathscr Z ,P)$ be a space of probability and $A_1,\dots,\in \mathscr Z$, then: $P (A_1)=P (A_1\cap A_2)+P (A_1\cap A_2^c)$ Dem: By ...
79 views

### Proof that if A and B are connected sets, $A \cup B$ is connected iff $(\bar A \cap B) \cup (A \cap \bar B) \neq \emptyset$

The problem says: Let $(X,\tau)$ be a topological space so that $A,B \subset X$ are connected sets in said space. For $M\subset X, \bar M$ refers to the closure of $M$. Prove that $A \cup B$ is ...
55 views

### A simple algorithm to find the inverse [duplicate]

Someone showed me such an algorithm. I'm going to explain his method with an example. Let's say we're looking for $$37^{-1}\in \mathbb Z/63\mathbb Z$$ First, we make the successive divisions of Euclid'...
1 vote
1k views

### Proof: center of a tree lies on the longest path

How can I make a proof of this property? I mean, given a weighted tree (with positive costs), how can I proof that the center of such a tree lies on the longest path? I read to the first answer of ...
65 views

### Show that the sequence $(1+N^2)/[N(1+N)]-5/6$ is positive and increasing

I have the function \begin{equation*} f(N)= \frac{1+N^2}{N(1+N)}-\frac{5}{6}, \end{equation*} where $N\in\mathbb{Z}_{++}/\{1, 2\}$. I want to show that $f(N)>0$ and that $f(N+1)>f(N)$ for ...
49 views

### Struggling with proving an elementary atomic equation. Please critique my proof.

I am learning how to prove and disprove atomic equations. I have put together a proof where I know how I want to prove the theorem but can't quite get it to flow correctly. Please critique my syntax, ...
3k views

### how to prove $G$ is an abelian group under $*$ (called the real numbers mod 1)

Let $G = \{x \in \mathbb{R}~|~0\leq x < 1\}$ and for $x,y \in G$ let $x*y$ be the fractional part of $x+y$ i.e $x*y = x + y - [x + y]$ where $[a]$ is the greatest integer less than or equal to $a$. ...
113 views

### Is there a visual proof for why this property of a matrix is true?

Let's say we have three equations written in their standard form: \begin{align} a_1x + b_1y + c_1 = 0 && (l_1) \\ a_2x + b_2y + c_2 = 0 && (l_2) \\ a_3x + b_3y + c_3 = 0 && (...
598 views

### Trigonometric Identities: Given that $2\cos(3a)=\cos(a)$ find $\cos(2a)$

Given that $2\cos(3a)=\cos(a)$ find $\cos(2a)$. $2\cos(3a)=\cos(a)$ I converted $\cos(2a)$ into $\cos^2(a)-\sin^2(a)$ Then I tried plugging in. I know this is not right, but I have no clue how to ...
69 views

### Proof that $\cos(x) = \cos\left(\frac{\pi}{2}\right)\implies x = \frac{\pi}{2} + k\pi$

I know that $$\cos(x) = \cos\left(\frac{\pi}{2}\right)$$ yields to $$(1)\qquad \qquad \qquad x = \frac{\pi}2 + 2k\pi$$ or $$(2)\qquad \qquad \qquad x = -\frac{\pi}2 + 2k\pi$$ which (as I found out in ...
2k views

### Is there a group theoretic proof that $(\mathbf Z/(p))^\times$ is cyclic?

Theorem: The group $(\mathbf Z/(p))^\times$ is cyclic for any prime $p$. Most proofs make use of the fact that for $r\geq 1$, there are at most $r$ solutions to the equation $x^r=1$ in $\mathbf Z/(p)$...
1k views

### Proving the relation between the Dirichlet eta function and the Riemann zeta function [closed]

The problem I am trying to solve is: I need to prove the relation between the Dirichlet eta function and the Riemann zeta function $\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$. But I have no clue ...
31 views

### Need hints (advice) to prove $(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$

I'm trying to prove this ( source : my uni's textbook says that it's trivial). $$(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$$ So far, I've managed to get ...
74 views

5k views

### Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.

I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$." This is my attempt of proving it:- Assume that $x=p/q$...
27k views

### prove there is no smallest positive rational number

How would I prove there is no smallest positive rational number? what is the best method to prove this statement?
I'm new to proof-writing in Real Analysis, and came across the following problem: Prove $\lim(a_nb_n)=\lim(a)\lim(b)$. Let $\lim(a), \lim(b)$ exist and $\lim(a)=L_a, \lim(b)=L_b$ The first thing I was ...
### Prove $1+2\sqrt3$ is not a rational number
How would I go about proving $1+2\sqrt 3$ is not a rational number assuming $\sqrt 3$ is not a rational? Would direct proof be the easiest? Total beginner here, any insight would be appreciated.