For questions about the formulation of a proof, not about the mathematics behind it.

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Proofs Using Tautologies

Let's say I want to formally prove a statement of the form $$p \implies q$$ So I do a bit of work,some re-arranging and eventually I arrive at a statement of the form $$p \implies p$$ which is a ...
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1answer
16 views

Proof Explaination: Show the set of measurable sets is closed under finite union

I have a proof of the above claim but I think there are some mistakes, I have highlighted them I hope someone could help figure out exactly what is wrong. Given $\omega$ an outer measure on set ...
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3answers
34 views

Verification of a proof in Measure Theory

Let $m$ be the Lebesgue measure on $\Bbb R$ and $f:\Bbb R\to [0,\infty)$ be a Lebesgue integrable function. Show that $\exists $ a measurable set $E\subset [0,\infty)$ such that $m(E)\neq m(f^{-1}(E)$ ...
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1answer
21 views

Polynomials and Lipschitz function

Let $f(x) = x^4 + 11x^2 + 9x -5$ and let $M > 0$. Show that f is a Lipschitz function on the interval $[-M, M]$ I honestly cannot figure out how to start this proof. Nothing similiar is in the ...
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0answers
9 views

Nonuniform Continuity Criteria with cos(1/x)

Use the nonuniform continuity criteron to show that $f(x) = cos\left(\frac{1}{x}\right)$ is not uniformly continuous on $(0, \infty)$ My proof: Let $(x_n)$ be defined by $x_n = \frac{1}{2n\pi}$ ...
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2answers
18 views

If $Mod(T_1 \cup T_2) = \emptyset$ then for some $\sigma$, $T_1 \vDash \sigma$ and $T_2 \vDash \neg \sigma$

Problem description: if $T_1$ and $T_2$ are theories such that $Mod(T_1 \cup T_2) = \emptyset$, then there is a $\sigma$ such that $T_1 \vDash \sigma$ and $T_2 \vDash \neg \sigma$. I don’t ...
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2answers
59 views

Is there a symbol for always less than (or just always?)

For e.g, the quotient of $\frac{1}{n}$, $q$, where $n \gt 1$, $q$ will always be less than $1$. $$\frac qn\le n$$ etc. I can't really write $\frac {q}{n} < n$, because whilst true, it doesn't ...
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0answers
29 views

How can this summation equation be proved by mathematical induction? [duplicate]

How can I prove this by mathematical induction? Sorry the wording is not very spacious, but it says "for all integers n." $$\sum_{j=1}^{n}{j^3}= \left(\frac{n(n+1)}2\right)^{2} \text{ for all ...
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3answers
96 views

Prove that $3^x$ divides $(3x)!\,\,\,\, \forall x \ge 1$ [closed]

Prove that $3^x$ divides $(3x)!\,\,\,\, \forall x \ge 1$ For example, $3$ divides $6 = 3!$
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2answers
37 views

Show that well-ordering is not a first-order property.

Problem description: Show that well-ordering is not a first-order notion. Suppose that $\Gamma$ axiomatizes the class of well-orderings. Add countably many constants $c_i$ and show that $\Gamma \cup ...
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3answers
38 views

Does proof proceed from left to right?

Very simple question: if we're asked to prove that $a=b$, do we start with $a$ and then find $b$ from $a$? Does going the other way around count as a formal proof? The exercise in question is this: ...
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3answers
32 views

How to prove that the g.c.d is equal to the prime factorization raised to the minimum of two powers

for the prime factorization of $a$ and $b$ as $$a = p_1^{\alpha_1}p_2^{\alpha_2}\cdots{p_t}^{\alpha_t}$$ and $$b = p_1^{\beta_1} p_2^{\beta_2} \cdots p_s^{\beta_s}.$$ I want to prove that $d = (a,b)$ ...
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1answer
37 views

show that $3^{1974} + 5^{1974} \equiv 0 \bmod 13$

show that $3^{1974} + 5^{1974} \equiv 0 \bmod 13$ My attempt with this question was to use Fermate Little's THM. But I do not understand how to properly use it for this question. Can some one show me ...
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1answer
18 views

Proof of equivalency in disjoint sets.

Prove, If A, B, C, and D are sets with |A|=|B| and |C|=|D| and if A and C are disjoint and B and D are disjoint, then |A ∪ C|= |B ∪ D|. Would I start this proof using the definition of disjoint ...
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3answers
136 views

Is it acceptable to use “But” in a proof that doesn't use contradiction?

I have recently read a lot of proofs that like to say "But..." right before the punchline. I feel that the word "But..." should be used if what follows is contradictory in some way, as in proofs by ...
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1answer
16 views

How to specify a function with flexible domain but same range?

As an example, I could be interested in functions that operate on $\mathbf{R}$ and $\mathbf{R}^2$. One way to say this is "all functions $f:\mathbf{R} \to \{0,1\}$ and all functions $f: \mathbf{R}^2 ...
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2answers
42 views
2
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1answer
58 views

Can I assume that random variables with exponential distribution are positive?

Let $(Y_n)$ be i.i.d random variables following exponential distribution with parameter $1$. Let $X_n=\min(Y_1,\dotsc, Y_n)$ Prove that $ X_n \xrightarrow{P} 0$ It's easy to prove that ...
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2answers
70 views

Proving some functor is adjoint to another. What to do with naturality condition?

Whenever I want to prove that some functor is (left/right) adjoint to another, I (mostly using hom-set definition) go on smoothly to prove the "isomorphism of the corresponding hom-sets", until it ...
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1answer
17 views

Inductive proof structure

To prove a statement about recursive series, is it correct to use an inductive proof structure showing that if $n = k$ and $n = k + 1$ are true then $n = k + 2$ holds true, and then prove the ...
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1answer
14 views

How to prove $|x-y|\le \delta\lor |x^2-y^2|\gt \epsilon$ with the following condition?

How to prove $\forall \epsilon\in \Bbb{R^+},\exists \delta\in\Bbb{R^+},\forall x\in\Bbb{R^+}, \forall y\in\Bbb{R^+}, |x-y|\le \delta\lor |x^2-y^2|\gt \epsilon$. My try: Pick ...
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0answers
18 views

How to prove $|n-m|\lt d\implies |f(n)-f(m)|\lt e$ for the following condition?

How to prove $\exists n\in\Bbb{N},\forall e\in \Bbb{R^+},\exists d\in\Bbb{R^+},\forall m\in\Bbb{N}, |n-m|\lt d\implies |f(n)-f(m)|\lt e$ if $f(x)=\lfloor \frac{n}{3}\rfloor$. I don't know how to ...
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2answers
30 views

Is it true that $X\iff Y$ is equivalent to $[X\land Y]\lor [\neg X \land \neg Y]$?

Is it true that $X\iff Y$ is equivalent to $[X\land Y]\lor [\neg X \land \neg Y]$? I don't see anything wrong with the statement. Could someone confirm?
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0answers
18 views

How to prove the following facts regarding the matrix

Let $X$ be a connected graph on $n$ vertices and $n$ edges. Let $Q$ be its edge incidence matrix.If $T\subset \{1,2 ,n\}$ with $|T|=n-1$ then $\det (Q[1,2 ,n-1\mid T])=^+_- 1$ if and only if the ...
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2answers
52 views

How to generalise this complex equation?

I am trying to generalise the statement for $n$ complex numbers: For any complex numbers $a,b,c$ with property $|a|=|b|=|c|=r\neq 0$. Prove $|\frac{ab+bc+ca}{a+b+c}|=r$ I proved this by showing ...
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2answers
31 views

Proof by Induction: Stuck

I feel as if this should be really really easy but is my brain getting there...? No... So here it is: Prove by induction that if $b$ is an odd number and $n$ is a positive integer, then $b^n$ is ...
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5answers
56 views

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$.

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor=\left\lfloor\frac{x}{n}\right\rfloor$. So we want to prove ...
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1answer
26 views

Proofing de Movire without Induction and in a neat way

The "usual way" gone for proving de Movire is via the road of induction. However this road get tiresome and thus wondered, if there were another way. However I came up with a proof that relies on ...
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1answer
37 views

Given sequences $(x_n)$ and $(y_n)$, define $(z_n)$ as $z_{2n-1} = x_n$ and $z_{2n} = y_n$. If $\lim x_n = \lim y_n = a$, so $\lim z_n = a$

Given sequences $(x_n)$ and $(y_n)$, define $(z_n)$ as $z_{2n-1} = x_n$ and $z_{2n} = y_n$. If $\lim x_n = \lim y_n = a$, so $\lim z_n = a$. I would like to know if my attempt and writing is ...
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2answers
31 views

Proof of non-uniform convergence of $x^n$

The wikipedia page on uniform convergence indicates that $f_{n}:[0,1]\to[0,1]$ with $f_{n}(x):=x^{n}$ converges pointwise but not uniformly to $$ f(x)= \begin{cases} 0,\quad x \in [0,1)\\ 1,\quad ...
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1answer
33 views

Least involved proof for continuos functions => uniform continuos functions on [a,b]

I have been looking at this proof in my textbook and seem to always get lost in its logic, its roughly 3 pages long. The proof is: If f is continuous on a closed interval [a,b], then f is uniformly ...
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2answers
31 views

Proving a Bunch of Statements

Been doing practices problems so... Consider the following statements: (a) Define what it means for a real number to be rational and for a real number to be irrational. Answer: I assume that this ...
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2answers
40 views

Proof Involving Irrational and Rational Numbers

Prove that if $xy$ is irrational then at least one of $x$ and $y$ is irrational. Here's what I did: Let $$r=xy$$ Assume $x\in\mathbb{Q}$ and $y\in\mathbb{I}$. Assume $r\in\mathbb{Q}$. Then ...
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1answer
44 views

Proof Fragment and Questions

Consider the following proof fragment. There exists an integer k such that $n=3k+1$. Then $n^2=(3k+1)^2=9k^2+6k+1=3(3k^2+2k)+1$ For each of the statements (a), (b), (c) below, answer the following. ...
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1answer
27 views

Show that $\mathbb{Q}(\sqrt{2})$ is a field.

Proof: Since $\mathbb{Q}$ is a field, then $\mathbb{Q}$ is a domain. (Theorem: if $R$ is a domain, then $R[x]$ is a field.) By the theorem, $\mathbb{Q}[x]$ is a field. So, letting $x = \sqrt{2}$, ...
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0answers
51 views

Mathematical proof using sequences

Let $\{x_n\}$ be a bounded sequence. a) Prove that there exists an $s$ such that for any $r > s$ there exists an $M ∈ \Bbb{N}$ such that for all $n ≥ M$ we have $x_n < r$. b) If s is a number as ...
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1answer
18 views

All triangles that have the same orthocenter and circumcircle have the same nine-point circle

True or false? Prove it. I guess it would help to figure out whether 2 triangles can have the same circumcenter or orthocenter and not be congruent. I have no clue how to figure this out. If they ...
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32 views

Mathematical Proof of sequences. Any help is appreciated

Let $\{x_n\}$ be a bounded sequence. a) Prove that there exists an s such that for any r > s there exists an M ∈ N such that for all n ≥ M we have $x_n$ < r. b) If s is a number as in a), then ...
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2answers
36 views

How to prove that $\limsup X_n \leq \sup\{X_n\}$?

How to prove that $\limsup X_n \leq \sup\{X_n\}$? I need to prove this and I don't know how to go about doing this. Thank you for any help you can provide.
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1answer
27 views

Induction proof (bitstring length)

Theorem : The number of bitstrings with the length $x$ that begin with $1$ and/or end with $0$ is $3 \times 2^{x-2}$. I know there are easier ways to prove this but I must figure out how to do it ...
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1answer
47 views

Induction Proof: If $B \subseteq A$, then $|B| \leq |A|$.

Prove by induction that if $A$ is a finite set and $B$ is a subset of $A$, then $|B|≤ |A|$. I can prove the base case with $n=0$ easily, but am stuck as to how to proceed from there.
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Theoretical Math Sequence Proof

Suppose that {xn} is a sequences such that every subsequence {xni} has a subsequence {xnmi} that converges to x. Show that {xn} is bounded. I tried to do a proof by contradiction but am not sure if ...
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6answers
33 views

Discrete math induction proof (divisibilty) [duplicate]

How to show that $10^n -(-1)^n$ is always divisible by $11$ through proof of induction?
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1answer
19 views

Infimum and supremum of finite ordered subsets

I am currently taking an introductory proofs course, and I have come across this problem. It's asking to prove the following: Let $S$ be an ordered set. Let $A$ be a non-empty finite subset. Then ...
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28 views

Recreational math dealing with twin primes

This is kinda recreational math with a goal in mind of progressing further toward a proof of the twin prime conjecture. Consider this: We start with a random prime: $109$ $3*109=327$ $327 ...
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2answers
27 views

Prove {$x|P(x) \land S(x)$}$ \cup $ {$x|P(x) \land \neg S(x)$} $= ${$x|P(x)$}

So far, I have only expanded the left hand side to [ {$x|P(x)$}$\cap ${$x|S(x)$} ] $\cup $ [ {$x|P(x)$ }$\cap ${$x|\neg S(x)$} ] and I'm not sure what to do next.
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2answers
24 views

Show $∀n≥3$, $2n^2+1 ≥ 5n$

I was able to prove the base case statement, where if you plug in $3$ for $n$ you get: $19 ≥ 15$. Next I supposed an arbitrary value $k$ where $k ≥ 3$ and $2k^2+1 ≥ 5k$. I know that next I need to ...
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0answers
31 views

How to prove this for number theory?

Let $n=p_1^{e1}p_2^{e2}\cdot \cdot \cdot p_k^{ek}.$ Then, $\phi(n)$= $n(1-$ $\frac{1}{p_1})$ (1- $\frac{1}{p_2})$$\cdot \cdot \cdot$ $(1- $ $\frac{1}{p_k})$ Hint: use the following If n has a prime ...
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1answer
16 views

Is this a sufficient proof for x-ε<a≤x?

The problem is: If y=Sup(S), show that, for each ε>0, there is a ∈ S such that x-ε < a ≤ x. Proof: Suppose x=Sup(S). Let S={n: n is a real number and n≤x}. Consider x-ε, where ε>0. Since x-ε is ...
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4answers
68 views

How to prove that there are no positive integer solutions $(x, y)$ to the equation $x^2 - y^2 = 1$

Prove the following: Theorem. There are no positive integer solutions $(x, y)$ to the equation $x^2-y^2=1$.