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

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1
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1answer
31 views

Verify my proof: for two positive natural numbers $x$,$y$ , if $ x + y = 2$ , then $ x = y = 1 $

Could someone verify my proof and my proof-writing? Proposition: for two positive natural numbers $x$,$y$ , if $ x + y = 2$ , then $ x = y = 1 $ Proof: Suppose $ y $ is any positive natural ...
1
vote
1answer
19 views

Convergence and metric - Proof?

Let $(x_n)$, $(y_n)$ be two sequences in a metric space $(P,d)$. Suppose $(x_n)$ converges to $x$ and $(y_n)$ converges to $y$. Prove that $\displaystyle\lim_{n \to \infty} d(x_n,y_n) = d(x,y)$ My ...
1
vote
1answer
37 views

Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$

Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$ My Approach Let $x_k$ be one element in a set of $n$ elements. $n-1\choose r-1$ $=$ the number of unique groups of $r$ containing ...
4
votes
2answers
206 views

A determinant problem

If $f(n)=\alpha^n+\beta^n$ and $$A=\left| \begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array} \right|$$ ...
1
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0answers
33 views

Simple proof that this sequence converges [verification]

This is a relatively simple problem. I'm just making sure I have the right idea here. I'd like to prove that the sequence $\displaystyle a_n = 1 + \frac{1}{n^{1/3}}$ converges. My proof is: We ...
2
votes
1answer
68 views

How to prove theorem using Euler's formula?

I'm having a great deal of trouble with this proof. "Prove $\cos θ + \cos 3θ + \cos 5θ + \cdots + \cos [(2n-1)θ] = \dfrac{\sin 2nθ}{2 \sin θ}$. Prove $\sin θ + \sin 3θ + \sin 5θ + \cdots + \sin ...
0
votes
2answers
50 views

Prove or disprove - If a divides b and b divides a does a=b

Prove or disprove: If a, b belong to the set of positive integers, and if a divides b and b divides a, then a=b. Does this hold if if a,b are not necessarily positive? Why or Why not? Here is what I ...
1
vote
1answer
18 views

For a linear function, the fiber of the output is the translate of the kernel by the input. (Trivial observation, proof needed.)

As you may already know, I am a newbie to linear algebra. I am supposed to prove that for every linear function between vector spaces, for every input, the fiber of the corresponding output equals the ...
0
votes
1answer
33 views

How can I show that a and b are odd in this contradiction proof?

Statement: suppose a,b belongs to Z (integers). If 4/(a^2+b^2) then a and b are not both odd. By proof of contradiction I assume that a and b are both odd. If a^2 and b^2 is odd then by definition a ...
1
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1answer
24 views

A collection of pairwise disjoint open intervals must be countable

Let $U$ be a collection of pairwise disjoint open intervals. That is, members of $U$ are open intervals in $\mathbb{R}$ and any two distinct members of $U$ are disjoint. Show that $U$ is countable. ...
0
votes
0answers
26 views

Can someone verify this proof? $\| x \|_2 \leq \| x \|_1$

Is this correct? $$\sum_{i \leq n} |x_i|^2 - \sum_{i,j \leq n} 2 |x_i||x_j| \leq \sum_{i \leq n} |x_i|^2 \implies \sum_{i \leq n} |x_i|^2 \leq \sum_{i \leq n} |x_i|^2 + \sum_{i,j \leq n} 2 |x_i||x_j| ...
0
votes
0answers
55 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
0
votes
1answer
56 views

Correctness of proof that $\lim_{n\to \infty}\sqrt n*c^n=0$

My proof is as follows: Assume $|c|\lt 1$ and $c$ can be written as 1/1+d for d>0 The definition of the mentioned limit is: For all $\epsilon>0$ there exists a natural number N s.t. for all n ...
0
votes
1answer
23 views

Proof that the set of irrational numbers is dense in the reals

The hint I was given was to simply prove that y=xz is irrational given that x is nonzero, x is rational and z is irrational. Here's how I did it: Claim: y=xz is irrational Proof: Assume $x\neq0$, x ...
2
votes
1answer
23 views

Correctness of proof that an ordered field S that has the supremum property also has the infimum property

First question I have is how would you describe the relationship between an ordered field and an ordered set and continue the proof by treating the field as a set? I want to say that right in the ...
2
votes
1answer
28 views

Algebra - proof verification involving permutation matrices

Theorem. Let $\textbf{P}$ be a permutation matrix corresponding to the permutation $\rho:\{1,2,\dots,n\}\to\{1,2,\dots,n\}$. Then $\textbf{P}^t=\textbf{P}^{-1}.$ Proof. First note the following ...
0
votes
1answer
37 views

What does the notation $H=\{ a | a^2=e \}$ mean? [on hold]

Is it true that the notation $H=\{ a | a^2=e \}$ means $H=\{a,a^2=e\}$?
1
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2answers
54 views

Is this proof by induction for a sum of odd squares correct?

Statement: $1^2 + 3^2 + 5^2 + ... + (2n - 1)^2 = (n/3)*(2n-1)*(2n+1)$ Proof by induction -Base case: when $n = 1$ $1^2 = 1/3 * (2 * 1 -1) * (2 * 1 +1) = 1$ $1=1$ hence statement holds for $n = 1$ ...
1
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2answers
51 views

Is this calculus proof I came up with sound?

We want to prove there every bounded sequence has a converging subsequence. Let $[l,u]$ be the interval to which we know $a_n$ is bounded.Let $\{a_n\}$ be the sequence and $[l_i,u_i]$ where $i$ is a ...
2
votes
2answers
46 views

Proving differentiable function is continuous.

To prove that if function has a derivative at a then it is continuous at $a$, my teacher did: \begin{align} & \|f(a+h)-f(a)\|=\|f(a+h)-f(a)+f'(a)~h-f'(a)~h\| \\[8pt] \leq {} & ...
0
votes
0answers
28 views

$f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$

I want to prove that if $f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$. This is my rather incomplete approach to ...
0
votes
2answers
39 views

Is this contradiction proof correct?

Statement : suppose $a,b$ belongs to $\mathbb{Z}$ (integers). If $4/ (a^2 + b^2)$ then $a$ and $b$ are not both odd. Proof by contradiction: Assume that if $4/(a^2 + b^2)$ then a and b are both odd. ...
0
votes
0answers
26 views

Measuring Unsigned Simple Functions

I was hoping that someone would be able to help me solve this problem regarding simple functions and their measure. This problem is coming straight from Introduction to Measure Theory by Terrence Tao. ...
0
votes
1answer
18 views

Groups and U27 double check

This is just a quick question. The Group U$_{27}$=$(1,2,3,5,7,11,13,17,19,23)$ right? Or am I just very wrong here?
1
vote
2answers
50 views

Proofread my work: Expressing generators of a cyclic group

The following question comes from Serge Lang's Undergraduate Algebra(pg. 26, 3rd edition). I just learnt the concept of groups and subgroups and I spent an hour or so on tackling part (b) of this ...
1
vote
1answer
34 views

Equivalence between two topological statements concerning the basis of a topology.

I need to show the following statement Let $\mathcal{B}\subset P(X)$ be a set of subsets of a set $X$, such that $\bigcup_{U\in \mathcal{B}}U =X$ then the following are equivalent $i)$ there ...
5
votes
1answer
48 views

Show that there exists $\xi\in [a,b]: f(\xi)=\xi$.

Let $a,b\in\mathbb{R},~a<b$ and consider $f\colon[a,b]\to [a,b]$ continuous. Show that $f$ has a fixed point. i.e. that there exists a $\xi\in [a,b]$ with $f(\xi)=\xi$. My idea is to ...
3
votes
2answers
55 views

Prove that an element of the basis is an element of the Kernel after linear transformation

Let $T:R^4\rightarrow R^4$ and basis $B=(v_1,v_2,v_3,v_4)$. $$T(v_1)+T(v_2)=T(v_3)\; \text{ and } \; T(v_1)+T(v_3)=T(v_2)$$ Prove that $v_1\in Ker(T)$ What I wrote is: $$T(v_1)=T(v_3)-T(v_2)\; ...
0
votes
2answers
22 views

Is this the correct way to prove by induction?

Prove by induction that $$1 + 3 + 5 + 7 + ... + (2n + 1) = (n+ 1)^2 $$ //for every n ∈ $\mathbb N$. $$1+2+3+...+n=\frac{n(n+1)}2$$ Proof: $$3+5+7+\ldots+(2n+1)=$$ ...
0
votes
1answer
13 views

Prove that $F'(x) = \sum_{n=1}^\infty F_n'(x)$ almost everywhere.

Suppose $F_n$ is a sequence of increasing non-negative right continuous functions on $[0,1]$ such that $\sup_n F_n(1) < \infty$. Let $F = \sum_{n=1}^\infty F_n$ and suppose that $F(1) < \infty$. ...
0
votes
1answer
46 views

Proof of FTA from Hatcher

This is the proof of the fundamental theorem of algebra (FTA) given in Hatcher's Algebraic Topology textbook (I have underlined the relevant part): Could someone explain why $r$ needs to be ...
1
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0answers
22 views

Has the abc conjuncture been proved by Shinichi Mochizuki? [duplicate]

I'd like to know is his proof was reviewed, and what exactly happened.
0
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2answers
25 views

Contrapositive proof question, is this a valid way

Definition: $a\in \Bbb Z$ is a perfect square if there is a $b\in\Bbb Z$ and $a = b^2$ To prove: if $m$ and $n$ are perfect squares, then $mn$ is a perfect square. I know that this can most easily ...
1
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0answers
43 views

Basis of $\mathbf{Q}[x]$

I wanna show that the binomials $\binom{x}{k}$ for $k=0,1,\ldots$ form a basis of the $\mathbf{Q}$-vector space $V=\mathbf{Q}[x]$. I can show that for fixed $m\in\mathbf{N}$ the $\binom{x}{k}$ ...
1
vote
2answers
57 views

how to prove $\sum_{i=1}^n i^k =\Theta(n^{k+1})$

we can say that if all $i$ s in the sum were equal to $n$ then the answer to the summation would be $n\cdot n^k$. So $n^{k+1}$ is the upper bound.so $\displaystyle\sum_{i=1}^n i^k=O(n^{k+1})$ For ...
0
votes
2answers
32 views

$AB*\text{adjoint}(BA)=I$

$AB*\text{adj}(BA)=I$ Prove: $1$. $|AB|=1$ $2$. $AB=BA$ As for $2$. what I have menage is $AB*AB^{-1}=AB^{-1}*AB=AB*\text{adj}$(BA)=I$ \rightarrow BA=AB$ How do I solve $1$. and is $2$. is ...
2
votes
1answer
30 views

Statistics - Show that $\hat{\theta}$ hat is a biased estimator of $\theta$

I'm asked to solve this exercise, but I can't manage to find something satisfying. Any help/hint would be much appreciated. Let $Y_1, Y_2,\dots, Y_n$ denote a random variable sample of size n from a ...
2
votes
1answer
17 views

Laws of equivalence

Need to proof using laws $$\lnot(p \land \lnot q) \lor q \equiv \lnot p \lor q$$ $\lnot(p \land \lnot q) \lor q$ $\equiv (\lnot p \lor \lnot(\lnot q)) \lor q\quad$ First De Morgan's law ...
1
vote
1answer
20 views

Star-Comb Lemma

I cannot understand that how can we apply Zorn's lemma here. What is the order set?
0
votes
1answer
17 views

Question about pointwise convergence of sequences in the box and product topologies.

Can someone please verify my proof or offer suggestions for improvement? I'm aware that there may be answers floating elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is ...
0
votes
0answers
22 views

Correctness of Proof that the limit of $\sqrt n\cdot c^n$ as $n$ tends to $\infty$ is $0$

The problem is: Given that $|c|<1$ prove that $\lim_{n\to\infty} \sqrt n \cdot c^n =0$. I am asked to use the comparison lemma and archimedean property to show convergence for sequence $\{1/\sqrt ...
0
votes
0answers
22 views

Correctness of Proof that the Archimedean Property of Reals is equivalent to lim $1/n$ as n tends to infinity

Here's what I have gotten so far: The Archimedean Property states 1) For every $\epsilon$ >0 there is a positive integer n s.t. $1/n$< $\epsilon$ and 2) For every positive number c there is a ...
3
votes
0answers
58 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
1
vote
3answers
54 views

Proof by contrapositive: $x^3 + 1$ is even if and only if $x$ is uneven

$x^3 + 1$ with $x \in \mathbb{Z}$ is even iff $x$ is uneven. I want to prove this using a proof by contrapositive, so this is my work: Assume that $x$ is even, so $x = 2k$ with $k \in ...
1
vote
3answers
66 views

$\mathbb{N} \times \mathbb{N}$ is countable?

Until this morning I was pretty sure that the answer was yes. It should have the same cardinality of $\mathbb{Q}$, which is countable. Besides the cartesian product of countable set should be ...
3
votes
2answers
31 views

We are given $f: X \rightarrow P(X)$, $f(x) = X\backslash\{x\}$, and $X$ is a set. Is the function injective, surjective, bijective?

I am working on this problem in a beginners set theory class. I believe the function is injective but not surjective, thus is it not bijective. We can show it is injective by letting $f(x) = f(x')$. ...
-1
votes
0answers
29 views

trigonometric equation (proof answer) [closed]

hi,all as you can see in the picture there are two parts that need to be proof. first is based on (b) and second based on (a) for the first equation, i already got the answer which is d3=2dm2. ...
1
vote
1answer
24 views

Local degree of local homeomorphism is $\pm 1$

Let $f:X\to Y$ be a local homeomorphism. I claim that local degree of $f$ is $\pm 1$. I was wondering if my proof is correct: Let $x\in f^{-1}(\{y\})$ , $U$ be a neighbourhood of $x$ and $V$ be a ...
0
votes
0answers
12 views

If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
0
votes
0answers
44 views

Proving that $3 = 9^{-1} \pmod{26}$

Prove that $3$ is the multiplicative inverse of $9 \pmod {26}$ $$\quad26\quad1\quad0\\2\quad9\quad0\quad1\\\;\;1\quad8\quad1\quad{-2}\\\quad\;1\quad-1\quad3$$ Hence $3$ is the multiplicative inverse ...