For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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1
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2answers
55 views

Proving $<$ is transitive on $\mathbb{Q}$.

I feel a little bit stupid asking this; I am asked to prove that, for all rational numbers if, x < y and y < z then x < z. I have said this; $ x + 0 < y $ $ x - z + z < y$ $ x - z ...
0
votes
2answers
38 views

Find $\lim_\limits{n\to\infty}\sin(\pi\sqrt[3]{n^3+1})$.

Find $\lim_\limits{n\to\infty}\sin(\pi\sqrt[3]{n^3+1})$. I am trying to find it using Taylor series. What I did so far is: $\sqrt[3]{z+1}=1+O(z)$ (I really can't tell when I should be done ...
1
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4answers
36 views

Proving that if $8\mid (n^2+2n)$ then $2\mid n$

Let $n\in \mathbb N$ prove that if $8\mid (n^2+2n)$ then $2\mid n$. From the given, there exsits $k\in \mathbb N$ such that $8k= (n^2+2n)$, take $k=1$, and we get $2\cdot 4 = n(n+2)$. Now my ...
1
vote
1answer
31 views

Doubt on presumably divergent series with primes

I am wondering if my reasoning is correct. I want to determine if the following series converges or not: \begin{equation} \sum_{n=1}^\infty\frac{1}{(\ln p_n)^2} \end{equation} where $p_n$ is the ...
1
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4answers
52 views

Proving $\forall x\in \mathbb R$, if $x>0$ then $(x+\frac 1 x \ge 2)$ [duplicate]

Prove $\forall x\in \mathbb R$, if $x>0$ then $(x+\frac 1 x \ge 2)$ I think a proof by contradiction is the easiest in this case, so we have: $\forall x\in \mathbb R :x>0\wedge \neg(x+\frac ...
5
votes
2answers
66 views

Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.

Let $f$ be differentiable on $\Bbb{R}$ and let $\lim_\limits{x\to \infty}f(x)=\lim_\limits{x\to -\infty}f(x)=0$. Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$. Attempt: If $f$ is constant we ...
1
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0answers
22 views

Proving that $\int_{\mathbb{R}} f \ d\mu = \frac{1}{N}\sum_{i=1}^N f(\lambda_i)$

I want to know if my proof is correct and if there is some easier way to prove this (you don't need to read all my proof, I'm accepting as answers another proofs, not just corrections of mine). ...
2
votes
1answer
66 views

Show $\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$ if $|z_1| <1$ and $|z_2| < 1$

Show $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$$ if $|z_1| <1$ and $|z_2| < 1$ Consider: $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right|^2$$ ...
0
votes
0answers
7 views

Proof verification (limit superior)

Could please somebody verify the proof? $x_n$ is a sequence of real numbers $\lim_{n \to 0} x_n \ne x$, show that $\exists \epsilon >0$, $\lim \sup_{n \to \infty} |x-x_n|>\epsilon$. Proof by ...
1
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1answer
20 views

Is the following subset of $\mathbb R \times \mathbb R$ with the indicated operation a group? Is it an abelian group?

$(a, b) * (c, d) = (ad + bc, bd),$ on the set $\{(x, y) \in \mathbb R \times \mathbb R: y \neq 0\}$. $(a, b) * (c, d) = (ad + bc, bd) = (cb + da, db) = (c, d) * (a, b).$ Commutativity holds. ...
5
votes
0answers
66 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all the positive integers $m$ such that both the ratios $$ \frac{2(5^m+5)}{3^m+1}, \frac{9^m+1}{5^m+5}$$ are integers. Attempt to a solution: If the ratios are both integers, than their ...
2
votes
1answer
36 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...
0
votes
0answers
11 views

Proof verification (limit inferior) is needed.

Could please somebody verify the following proof? We have a sequence of real numbers $x_n$ such that $x_n \ge f(t- \epsilon)-p(\epsilon,n)$, $p(\epsilon,n)$ $\forall \epsilon$ and $\forall n$, ...
4
votes
3answers
64 views

Is my proof that $\lim_{x\rightarrow 0} x\sin\frac{1}{x}=0$ correct?

I tried to solve this limit: $$\lim_{x\rightarrow 0} x\sin\frac{1}{x}$$ And I arrived at the answer that $\lim_{x\rightarrow 0} x\sin\frac{1}{x}=0$. Is my solution correct? $\lim_{x\rightarrow 0} ...
0
votes
1answer
32 views

Prove that each of the following sets, with the indicated operation, is an abelian group

$1.$ $x * y = x + y + k$ ($k$ a fixed constant), on the set $\mathbb R$ of the real numbers. $x * y = x + y + k = y + x + k = y * x.$ Commutativity holds. $(x * y) * z = (x + y + k) * z = (x + y + ...
0
votes
0answers
13 views

Comparison of pseudomonotone definitions

Are the intersections $\bigg(\cdot\bigg) \bigcap[x,y]$ necessary for the terms on page 3? Or could the proof follow by dropping the $[x,y]$? See paper. Thanks
3
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0answers
46 views

Every tree has two leaves. Is my proof ok?

A tree is a connected acyclic graph. A leaf is a vertex of degree one. The distance $d(u,v)$ between two vertices $u$ and $v$ of a graph is the length of the shortest path from $u$ to $v$. Theorem. ...
3
votes
1answer
41 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
1
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1answer
47 views

Is there something wrong with this proof of the cancellation rule?

I was helping a friend of mine with math homework. We were dealing with real numbers, and had to prove the cancellation rule; that is $$a,b,c, \in \mathbb{R} \land a+c=b+c \implies a=c$$ He had a ...
0
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0answers
22 views

Intro. analysis - proof that $x \in N_{\epsilon}a$

Define: $$N_{\epsilon}a=\{x: |x-a|< \epsilon\}$$ Show that if $\epsilon$>0 and $|x-a|<\epsilon$ then $x \in N_{\epsilon}a$ I first note that $N_{\epsilon}a =(a-\epsilon,a+\epsilon)$ then I ...
-1
votes
0answers
27 views

Quadratic form - non-degenerate

(The order of a quadratic form is defined to be the order of the matrix $A$) Definition: $Q(x_1, x_2, \dots , x_n)$ is called non-degenerate $\Leftrightarrow (a) $A=$invertible (b) At each $v \in ...
1
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0answers
66 views

Determine all $t\in\mathbb{R}$ for which $A_t$ is diagonalizable.

I have this matrix: $$A_t=\begin{pmatrix}\phantom{-}2+t&\phantom{-}4&\phantom{-}2+t&\phantom{-}2+t\\\phantom{-}t-2&\phantom{-}0& -6+t& ...
0
votes
1answer
23 views

proof detail concerning bijection between a set and its power set

Theorem: If $X$ is a set, then $X$ is not equivalent to its power set. Proof: suppose for a contradiction that $f:X\to P(X)$ is a bijection. Define $B:=\{x \in X, x\not\in f(x)\}$. Because $f$ is ...
0
votes
0answers
29 views

Riemann integrable function to the power of $p \in [1,\infty[$ is R-int. again.

It is stated that it is sufficient to prove Riemann-integrability of $|f|^p$ for $0 \leq f \leq 1$. $(f,\psi,\varphi:[a,b] \rightarrow \mathbb R)$. $\checkmark$ For any $\varepsilon >0$ there ...
5
votes
5answers
55 views

Demonstration of sum of powers of $2$ [duplicate]

Theorem : For every natural number $p$: $$\sum^p_{i=0} 2^i = 2^{p+1}-1$$ I trieed to demonstrate the theorem using induction Demonstration : $1)$ If we have $p=0$ then we get $2^0=2^{0+1}-1$ that is ...
1
vote
2answers
38 views

Prove/disprove $A\cap B=A\cap C $ for every $A$ $\iff B=C$

Let $A,B,C$ be sets, prove/disprove: $A\cap B=A\cap C $ for every $A$ $\iff B=C$ I think it's wrong, choose $A=\{1,2\}, B=\{2,3\}, C=\{2,4\}$ so $A\cap B=A\cap C$ but $B\neq C$ Although it's a ...
1
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2answers
32 views

Proving linear dependency for two vector groups

The question: Let V be a vector space over $\mathbb{R}$. Let $S = \{v,u,w\}$ be a group of 3 vectors in V. Let T be defined as $T = \{v, v + u, v + u + 2w \}$. Prove that if S is linearly dependent, ...
0
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1answer
27 views

Simple inequality proof in analysis

Just need verification on whether my proof is valid. I couldn't find a straightforward way to prove this inequality directly, so I tried a proof by contradiction instead. The question: Let $a, b \in ...
1
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0answers
25 views

Proving with a given definition that if $|A|=|B|$ then $A,B$ are equivalent (with induction but without using the induction hypothesis)

Let $A,B$ be finite sets, we'll say the sets are equivalent if $|A\setminus B|=|B\setminus A|$. Prove with the above definition that if $|A|=|B|$ then $A,B$ are equivalent. Suppose ...
0
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3answers
33 views

Proving if $p|ab$ then $p|a\vee p|b$, then $p$ is prime

Let $1\neq p\in \mathbb N$ such that $\forall a,b \in \mathbb N$ if $p|ab$ then $p|a\vee p|b$. Prove that $p$ is prime. My attempt, proof by contradiction: Suppose $p$ isn't prime, then ...
1
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2answers
35 views

Proving Pascal's identity

So I came across Pascal's identity: Prove that for any fixed $r\geq 1$, and all $n\geq r$, $$ \binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}. $$ I know you can use basic algebra or even an inductive ...
2
votes
3answers
54 views

Prove that $0 < 1$. Prove that $ab = 0 \implies a = 0$ or $b = 0$.

Proof: There exists $a = 0$ (For every $b$, an element of the set of positive numbers, such that: $b > a$) $$a + b > 0 \implies b > 0 \implies a < b.$$ Thus, we have shown that $0 < ...
1
vote
1answer
49 views

Is the solution to this elementary number theory problem correct?

Problem: A natural number $n$ is called nice if the following properties hold: • The expression is made ​​up of 4 decimal digits; • the first and third digits of $n$ are equal; • the second and ...
0
votes
3answers
38 views

Prove that if $l$ is a line in the classical euclidean plane, then there is a point $p$ that lies on $l$

Suppose that $\mathbb{P}$ is a Classical Euclidean Plane (satisfies all five of Euclid's postulates). Can you prove that if $L$ is a line in $\mathbb{P}$, then there is at least one point $p$ in ...
1
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1answer
20 views

Prove that for all $m$, there exist some $k$, such that $(m-n)^2 > m^2$ for all $n>k$

I have a problem where I need to prove: $\forall m \in \mathbb{N}:\exists m \in \mathbb{N} ∋(m−n)^2>m^2~∀n>k$ My thought was since it is only "there exists some k.." can I not say: if $k = ...
2
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0answers
29 views

Prove that $U_1\cup U_2$ is a subspace of $V$ $\iff$ $U_1\subseteq U_2$ or $U_2\subseteq U_1$ $\triangle$

Let $V$ be a vector space over some field. Let $U_1$ be a subspace of $V$. Let $U_2$ be a subspace of $V$. Prove that $U_1\cup U_2$ is a subspace of $V$ is equivalent to $U_1\subseteq U_2$ or ...
0
votes
1answer
28 views

Rectangles in one dimension

I have to prove the following proposition : Show that the intesection of two rectangles in $\mathbb{R}^{n}$ is either the vaccum or is another rectangle. My attempt: I one is embeded in the other ...
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0answers
36 views

Example: How to find inverse Laplace Transform by integral of the function (5.2-29)

This is just a demonstration on how to solve the following type of problem. Find $\mathcal{L}^{-1}\{\frac{54}{s^3(s-3)}\}$ by the given method: $$\mathcal{L}\{ ...
1
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2answers
44 views

Need help understanding Fibonacci Fast Doubling Proof

From this website, http://www.nayuki.io/page/fast-fibonacci-algorithms (fast doubling proof close to the bottom of the page). I have understood the proof for the most part but I am struggling to see ...
3
votes
4answers
63 views

Prove $\lim_{x\to2} (x^4 - 2x^3 + x + 3) = 5$ using Epsilon Delta

Prove $\lim_{x\to2} (x^4 - 2x^3 + x + 3) = 5$ using Epsilon Delta I am having difficulty finding the $\delta$ value. Here is what I have done so far: What I want to show: $$\forall \epsilon > ...
3
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1answer
39 views

Verifying a Vector Space Via Given Axioms

Let $X$ be the collection of all sequences $\{\alpha_n\}_{n=1}^{\infty}$ of scalars from $\mathbb{K}$ such that $\alpha_n=0$ for all but a finite number of values of $n$. Define addition and scalar ...
2
votes
1answer
43 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
1
vote
1answer
28 views

Proving that something diverges to infinity.

So I'm trying to prove that the sum of 1/(2k+1) diverges to infinity. I thought about doing a comparison test with the harmonic series 1/k and multiplying the harmonic series by (1/3) so it is (1/3k). ...
2
votes
0answers
33 views

Fiber product of non-abelian groups.

I am trying to understand whether surjectivity is needed for a fiber product of non-abelian groups to exist. I seem to have checked that the usual construction works for groups without any ...
1
vote
2answers
56 views

How to get to $5^3 \geq n^3$ in the proof by contradiction?

This is the same problem asked here. - Next step to take to reach the contradiction? Here is it again. I understand the solution - how you want to get to the fact 100 divides n^2 and then go ...
2
votes
2answers
31 views
+50

Homology groups of orientable surfaces.

Edit: I have a proof here but when I spoke last with my professor, she told me something was close, but not quite. Can someone help me patch this proof? I've been trying to get this down for quite a ...
1
vote
0answers
23 views

For what positive integer values $b,d$ does $(b^2-d)\mid(b^2-1)?$ hold?

I am curious about the answer to the following questions: And hope that you can help me For what positive integer values $b, d$ does $$(b^2-d)|(b^2-1)?$$ hold? Is it correct that the only ...
0
votes
1answer
22 views

Two example statements meant to demonstrate the importance of quantifier order don't appear to do so [duplicate]

In a book1 I have encountered the following: To check your understanding of [the importance of quantifier order], consider the following two statements. One is true, and the other is false. Which ...
0
votes
2answers
38 views

Prove that $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0})] = 0 \Rightarrow \lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$

Prove: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0})] = 0 \Rightarrow \lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ Proof: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ ...
2
votes
1answer
45 views

Proof Verification (Set Theory)

Let $S$ be a set with $N$ elements and let $A_1,\dots ,A_{101}$ be $101$ (possibly non disjoint) subsets of $S$ with the following properties: a) each element of $S$ belongs to at least one of these ...