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Proof that there are infinitely many primes congruent to 3 modulo 4

Does this work? This is a modification of Euclid’s proof. Let $p_1,p_2,…,p_k$ be the only 3 mod 4 primes. Consider two cases : (I) $k$ is odd Then notice $p_1p_2…p_k+4$ is a 3 mod 4 number, and in ...
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How to prove the product of first n consecutive odd numbers is a square less than another square?

You might note that the difference sequence of the sequence of squares is exactly the odd numbers. Here's the sequence of squares: $$0, 1, 4, 9, 16, 25, 36, \ldots$$ Here's the sequence of the ...
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Exercise 7(a), Section 31 of Munkres’ Topology

Edit: I am expanding this instead of commenting since stack exchange doesn’t seem to like so many comments: Approach 2 does not work. In order to use your cited theorem, you would need the following 1 ...
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Why Can't $\delta$ depend on $x$ in $\delta-\epsilon$ Proofs

You are trying to find a single $\delta$ value ahead of time that works for many cases. The cases are where $0 < |x-a| < \delta$, and the definition involves trying to find one delta $\delta$ ...
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How to prove the product of first n consecutive odd numbers is a square less than another square?

Note that any product of two distinct odd numbers or two distinct even numbers can be written as a difference of squares. Let $p<q$ be both odd or both even. Then $(q-p)/2$ and $(q+p)/2$ are ...
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Is this valid to prove that $2 \ \vert \ x^3 \Rightarrow 2 \ \vert \ x$?

Suppose $2 \mid x^3$. Assume $2 \nmid x$, then $\exists r \in \mathbb{N}$ such that $x = 2r + 1$. Then $$x^3 = (2r + 1)^3$$ $$=8r^3 + 12r^2 + 18r + 1$$ Since $2 \mid x^3$, then it follows that $2 \mid ...
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How to prove $\forall x\; (\phi (x)\land \psi (x) ) \rightarrow \forall x\; \psi (x)$ without using the completeness theorem?

Here's a direct proof using the system in your book. $(\forall x\, (\phi(x)\land \psi(x)))\to (\phi(x)\land \psi(x))\quad$ (Q1, substituting $x$ for $x$) $(\forall x\, (\phi(x)\land \psi(x)))\to \psi(...
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Proving that $\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$.

Seems like it's on the right track. But one way to streamline the presentation, and avoid the case work, is to write the sign as $\epsilon_{I,I^c}=\text{sgn}(I,I^c)$, the sign of the permutation $(1,\...
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Is this valid to prove that $2 \ \vert \ x^3 \Rightarrow 2 \ \vert \ x$?

There are a couple issues: You didn't really show that $2$ does not divide $x^3$. Rather, you showed that, for all $k\in\Bbb Z$, $x^3\neq 2(4k^3)$. It's not clear why $x^3$ must be of the form $2(4k^...
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Is this valid to prove that $2 \ \vert \ x^3 \Rightarrow 2 \ \vert \ x$?

This method is not valid. You showed that $2(4k^3)\neq x^3$ for all $k\in\mathbb Z$ correctly, but but to show that $2\nmid x^3$, you need to show that $2n\neq x^3$ for all integers $n$, not just ...
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Is the interior of the closure of a set equal to the interior of that set?

Think about the set $\mathbb R$$-${$0$}
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Suppose $\sum_{k=1}^{\infty}{a_{ 2k+1}}$ and $\sum_{k=1}^{\infty}{a_{2k}}$ converge. Does $\sum_{n=1}^{\infty}{a_{n}}$ converge?

Yes it converges. You can simply consider $\sum_{n\geq 1}a_n$ and split it into its odd and even coefficients which gives you $$\sum_{n\geq 1}a_n=\sum_{k\geq 0}a_{2k+1}+\sum_{k\geq 1}a_{2k}.$$ Given ...
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Exercise 12, Section 26 of Munkres’ Topology

The idea I think is the following: $X$ is compact if and only if any family $\{C_\alpha\}$ of closed sets of $X$ with the f.i.p. Property admits $\cap_\alpha C_\alpha\neq \emptyset$. Take a family of ...
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Show that the set of $n^{th}$ roots of unity form a subgroup under the group $S^{1}$

Are you familiar with modular arithmetic? $k_{1}+k_{2}= kn+r $ for some $0\leq r\leq n-1$ and $k$ is some integer. Hence $$c_{1}c_{2} = e^\frac{2 \pi k_{1} i}{n}e^\frac{2 \pi k_{2} i}{n} = e^\frac{2 \...
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Questions about Proof by Induction

Fundamentally, a proof by induction is using the well-ordering property of the natural numbers, i.e., the fact that every non-empty set of natural numbers has a least element. And it can always be ...
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Questions about Proof by Induction

I'm not sure I can answer ##3 and 4 authoritatively. But I've never heard of proof by induction being used for n other than integers. And I've never heard of a circumstance where proof by ...
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Questions about Proof by Induction

(1) The base case is not necessarily $n=1$. One might want to prove a statement is true for all numbers $n$ greater than or equal to a certain number $m$. To understand the importance of the base case ...
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Proving that $\frac{n^2-1}{n^2}\left(\frac{1}{p}+\frac{a}{b}\right)$, with conditions on integers $a$, $b$, $n$, and prime $p$, is never an integer

Here is a counterexample: $\frac{22² - 1}{22²} \left( \frac{1}{23} + \frac{925}{21} \right) = 44 \in \mathbb{Z}$.
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Let $\mathcal{A}= \{A_n = \left(0, \frac{n}{n+1} \right) \ | \ n \in \mathbb{N} \}$. Prove that $\bigcup \mathcal{A}\subseteq (0,1)$.

The sequence $A_n=(0,\frac{n}{n+1})$ is increasing. $\bigcup_n A_n=\underset{n\to\infty}{\text{lim}}A_n=\underset{n\to\infty}{\text{lim}}\left(0,\frac{n}{n+1}\right)=\underset{n\to\infty}{\text{lim}}\...
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Let $x \in (0,1).$ How to show that $x \in \bigcup \left(0, \frac{n}{n+1}\right)$?

Since $\frac{n}{n+1}$ is an increasing function with argument $n$ on $\mathbb{N}^+$, then we have \begin{align*} \bigcup_{n=1}\left(0,\frac{n}{n+1}\right)&=\left(0,\lim_{n\to \infty}\frac{n}{n+1}\...
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Let $x \in (0,1).$ How to show that $x \in \bigcup \left(0, \frac{n}{n+1}\right)$?

In order to show that $x$ is in the union; $$\displaystyle\bigcup_{n=1}^{\infty}{\left(0,\frac{n}{n+1}\right)},$$ you'll have to show that $x$ is in one of the intervals of the form; $$\left(0,\frac{n}...
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Let $x \in (0,1).$ How to show that $x \in \bigcup \left(0, \frac{n}{n+1}\right)$?

Since $$ \mathop {\sup }\limits_{n \in \mathbb N} \left( {\frac{n}{{n + 1}}} \right) = 1 $$ it follows that for each $x \in (0,1)$, from Archimedean Property, there is $n_1 \in \mathbb N$ so that $ x &...
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Proving the function $f(x) = x^2 + ax + b$ is not injective. Does my proof make sense?

Correct ideas are in your proof. Your logical presentation is not right, though. You shouldn't start the proof by saying "Let $m, n \in \mathbb R$ be such that $f(m) = f(n)$." Rather, you ...
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Let $\Gamma \neq \emptyset$. Also $\Gamma \subseteq \Omega$. Show that $\bigcap \Omega \subseteq \bigcap \Gamma.$

A possibly better way is as follows. Proof. Since $$ \begin{array}{rcll} x\in \bigcap\Omega&\Leftrightarrow&\text{for all }X\in\Omega\text{, we have }x\in X&\text{by the definition of }\...
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How to write maths

Singular can be the instance while the plural can be the iteration or collection of instances although it would be contextually dependent. Multiplication - $x(n)$ Multiplications - $\int_{x(n)}$ In ...
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Prove: if Γ ⊢ L : σ, then Γ is a λ2-context.

Definition 3.4.4 defines $\lambda2$-contexts. Figure 3.1 summarizes the derivation rules of $\lambda2$, i.e. when $- \vdash - : -$ holds (would have been clearer had they have written $\vdash_{\...
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Range of a quadratic equation

In method 2, we can not use symbol $\Leftrightarrow$ because $\frac{x^2-4}{x-2}=y$ is not same as $x^2-4=xy-2y$ because the later expression is true for $x=2$ but the former is not true for $x=2$. To ...
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How do I prove formally that for all natural numbers $a\cdot S(c)=b\cdot S(c)\implies a\cdot c=b\cdot c$

Let's prove some preliminar properties. I'm assuming commutativity, associativity and the distributive property (which are true by the way). I need some other useful properties (you can read only the ...
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How do I prove formally that for all natural numbers $a\cdot S(c)=b\cdot S(c)\implies a\cdot c=b\cdot c$

Mathematical induction is given as follows: $1.\,\,P(0)\\2.\,\,(\forall n \in \mathbb{N})[P(n) \implies P(S(n))]\\3.\therefore (\forall n \in \mathbb{N})[P(n)]$ For this proof we will have to define P(...
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Prove that $n$ is a composite number if $n + 11$ and $n + 7$ are prime for all natural numbers $n > 3$

All primes greater than $3$ are of the form $6k\pm 1$. Plainly $n+11>n+7>3$, so if $n+11,n+7$ are prime, they have the form $6k\pm 1$. Since $11=6\times 2 -1$ and $7=6\times 1 +1$, it follows ...
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Prove that $n$ is a composite number if $n + 11$ and $n + 7$ are prime for all natural numbers $n > 3$

Clearly $3$ divides only one of $n+8$, $n+9$ or $n+10$. Since $n+7$ is prime, $3$ doesn't divide $n+10$. Similarly since $n+11$ is a prime, $3$ doesn't divide $n+8$. Thus $3$ must divide $n+9$. But ...
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For all natural numbers $n$, if $n$ is odd, then $\sqrt{15^n}$ is irrational.

I'd start by saying that if $n$ is the odd natural numbers then it can be represented as: $$n=2k+1\quad\forall k\in\mathbb{N}_0$$ Secondly, $$15=5\cdot3\rightarrow \sqrt{15^n}=5^{n/2}3^{n/2}$$ now ...
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For all natural numbers $n$, if $n$ is odd, then $\sqrt{15^n}$ is irrational.

If $n=2k+1$ for some $k \in \mathbb{Z}$, then $$ \sqrt{15^n} = 15^{(2k+1)/2} = 15^k \sqrt{15}. $$ So if $\sqrt{15^n} \in \mathbb{Q}$, then $\sqrt{15} \in \mathbb{Q}$, as well. Can you show $\sqrt{15}$...
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Computing $\omega^n=\omega\cdots \omega$ ($n$ fold product), where $\omega=\sum\limits_{k=0}^{n}dx_idy_i$

Your proof seems correct. I feel however that proceding by induction is not really necessary, as you can appeal to combinatorics. Let me explain better what I mean: to ease notation, let $\eta_i=\...
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Prove $|\sum_{j=0}^n \frac{5^j}{c^j}-\frac{c}{c-5}|\leq\frac{(5^{n+1})}{|c|^n(|c|-5)}$

$\sum_{j=0}^n \frac{5^j}{c^j}=\frac {1-(5/c)^{n+1}} {1-5/c}$ by the formula for geomertic sum with common ratio $5/c$. So the left side of the inequality is $|\frac {1-(5/c)^{n+1}} {1-5/c}-\frac c {...
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Prove $\frac{((1+(|k|)^{1/8})^{4n}}{n(n-1)(2n-1)(2n-3)}\leq \frac{(4n-1)(4n-3)(4n-5)(4n-7)}{630}|k|$

At $k=0$, the inequality doesn't hold, as the left side is positive and the right side is nonzero. Now, for $k\neq 0$, the inequality is the same as $$\frac{(1+x)^{4n}}{x^8}\leq \frac{n(n-1)(2n-1)(2n-...
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proof that if $A$ is open in $(X,d_X)$ then the subset $G $ of $A$ is open in $(A, d_A)$ if and only if it is open in $(X, d_X)$

The key observation is that a ball in $A$ could be represented as $B_A(x, r)=A\cap B_X(x, r)$. This can imply that any open set in $A$ could be represented as $A\cap U$, where $U$ is open in $X$.
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Is the following characterisation of measurable functions true?

Not true without the additional assumption of completeness of the space. Let $X=\mathbb R, \mathcal S =$ Borel sigma algebra and $\mu$ be Lebesgue measure. Let $C$ be the Cantor set. There exists a ...
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Prove that for all natural numbers $\neg\big(S(a)+b=a\big)$

We will use the following auxiliary results. Lemma 1 $\forall b\forall a\big(a+S(b)=S(a)+b\big)$. Proof We will proceed by induction on $b$. Base case. Assume that $b = 0$. We need to show that $\...
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Prove that $(A \cup B) \triangle A = A \setminus B$

Since \begin{align*} (A\cup B)\triangle A&=((A\cup B)\setminus A)\cup(A\setminus (A\cup B))\\ &=(B\setminus A)\cup\varnothing\\ &=B\setminus A, \end{align*} then the equality should be $(A\...
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if $c+q\in \Bbb R$and $cq\in \Bbb R$ then $|c-4q|^2=|c|^2-8|c||q|+16|q|^2$?

$\underline{\text{Question 1}}$. if $c+q\in \Bbb R$and $cq\in \Bbb R$ then $|c-4q|^2=|c|^2-8|c||q|+16|q|^2$? Assume that $(a + ib) = c, (x + iy) = q.$ Since $c + q \in \Bbb{R},$ you have that $y = -...
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Prove there exsits some $k\in \Bbb R$ such that $k^4+4t^2+5\lt0$.

The statement in the question, for which it is asked to determine whether it is true or false, is of the form $$P \implies Q,$$ where $P,Q$ are two (simpler) statements. Thus if it happens that $P$ is ...
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Prove $a^{1/k}+b^{1/k}\gt (a+b)^{1/k}$

Firstly whenever proving any sort of algebraic (in)equality I would try to isolate a useful property to simplify the equation. In this case, $$ (a^{\frac{1}{k}}+b^{\frac{1}{k}})^{k} > [(a+b)^{\frac{...
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Prove that $ 2/1!+4/3!+6/5!+8/7!+\dots = e$

Let $f(x)= x \sin(x)$. Then $$ f(x) = \frac{x^2}{1!} - \frac{x^4}{3!} + \frac{x^6}{5!} -\frac{x^8}{7!} + \cdots $$ and so $$ f'(x) = \frac{2x}{1!} - \frac{4x^3}{3!} + \frac{6x^5}{5!} -\frac{8x^7}{7!} +...
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The set of $d\vec{x}_I$ with $I$ increasing is a basis of the vector space $\Lambda^k(\mathbb{R}^n)^*$ of alternating multilinear functions

You can shorten and make the proof for spanning set a lot cleaner by considering only basis vectors. I'll denote $\Bbb{R}^{n}$ by $V$ for easier typing. Let $T\in \Lambda^{k}(V^{*})$ Define $T_{I}=T(...
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Exercise 1, Section 18 of Munkres’ Topology

Let $f$ be continuous in the $\varepsilon-\delta$ sense, and let $V$ be an open subset of $\mathbb{R}$. Let $x_{0} \in f^{-1}(V)$. Then $V$ contains an interval of the form $\left(f\left(x_{0}\right)-\...
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Exercise 4, Section 18 of Munkres Topology

$f$ and $g$ are clearly injective. They are also continuous as each of their components is continuous so the maps $f^{\prime}: X \rightarrow X \times\left\{y_{0}\right\}$ and $g^{\prime}: Y \...
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Exercise 12, Section 30 of Munkres’ Topology

Both parts of your proof are correct. However, since you included a proof-writing tag in your question, I'd like to add a quick comment about style! You use a lot of symbols, and that makes your proof ...
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Proof Verification for $e^x$ is not uniformly continuous on R

The $\delta$ you choose depends on the $\varepsilon$ so you can't take $\varepsilon$ in terms of $\delta$. However, if you fix $\varepsilon >0$ and suppose that there exists $\delta>0$ such that ...
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Proof Verification for $e^x$ is not uniformly continuous on R

Choose the sequence $\log(n+1)$ and $\log n$ and this two are parallel sequences but $e^{\log(n+1)}$ and $e^{\log n}$, not to be a parallel sequences, so that's why $e^x$ is not uniformly continuous.
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