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

learn more… | top users | synonyms

0
votes
2answers
29 views

Prove or disprove the following asymptotic relations

$P(x) = 2^x$ Prove or disprove that $p(n^3 + 4) \in O\left(p\left(n^3\right)\right)$ $2^{(n^3 + 4)} \in O(2^{n^3})$ $\lim_{n \rightarrow \infty} \space \frac{2^{n^3 + 4}}{2^{n^3}}$ using ...
1
vote
1answer
49 views

Let $(s_n)$ be a sequence in $\mathbb{R}$. Prove $\lim_{n \to \infty}s_n=0$ if and only if $\lim_{n \to \infty}|s_n|=0$

First, assume that $\lim s_n=0.$ This implies that for any given $\epsilon > 0$, $\exists$ an $N$ such that for $n>N,|s_n-0|< \epsilon$. $|s_n-0|=|s_n|<\epsilon$ and $|s_n|=||s_n|-0| ...
3
votes
0answers
57 views

If $A$ and $B$ are arbitrary $m\times n$ matrices, show that $^t(A+B)= {}^tA+{}^tB$?

I'm reading Lang's: Introduction to Linear Algebra. There is this exercise: If $A$ and $B$ are arbitrary $m\times n$ matrices, show that $^t(A+B)= {}^tA+{}^tB$ I did the following: ...
1
vote
1answer
39 views

Is this a correct proof that $\tau(n)$ is eventually smaller than any $n^{\delta}$, $\delta>0$?

Is this a correct proof that $\tau(n)$ is eventually smaller than any $n^{\delta}$, $\delta>0$? I'm not even sure about the statement, let alone the proof. Let's first proof this result: $\tau(n) ...
0
votes
2answers
79 views

prove $\langle f(x),f'(x)\rangle = 0$

Let $f: R \to R^n$ be a differentiable function such that $\forall x \quad||f(x)|| = 1$ prove that $\forall x \quad \langle f(x),f'(x)\rangle = 0$ i thought of the following proof but not sure it ...
0
votes
1answer
46 views

Which of the following sets are countable?

a) $[0,1] \cap \mathbb Q$ b) $P(\mathbb Q)$ c) $\mathbb R \setminus \mathbb Q $ d) $\{(a, b) ∈ \mathbb R\times\mathbb R | a, b \in\mathbb N\}$ I answered a) and d) a) any intersection between ...
0
votes
5answers
47 views

Is $F:R\to[0, ∞)$ where $F(x) = e^x$ a bijection?

Is $F:R\to[0, ∞)$ where $F(x) = e^x$ a bijection? For a function to be surjective, the function must hit all elements belonging to the CODOMAIN (Which is $[0,∞)$ right?) or does it simply have to ...
0
votes
0answers
19 views

Partial sum of a product over an arbitrary sequence.

Below is an equation a friend showed me, but was unable to prove. After struggling with it for a bit I was unable to as well. After failing to show this for, say, N=2 Im pretty sure the equation is ...
0
votes
1answer
41 views

Which of the following are bijections?

• $f : \mathbb{Z} → \mathbb{Z} \\ f(x) = x^5 - 3$ • $g : \mathbb{R} → \mathbb{R} \\ g(x) = x^5 - 3$ • $h : \mathbb{Q} → \mathbb{Q} \\ h(x) = x^5 - 3$ • $F : \mathbb{R} → [0, ∞) \\ F(x) = e^x$ ...
2
votes
3answers
24 views

Let $A = \mathbb{Z}$, $B = [−1, \pi]$, $C = (2, 7)$. List all elements of $A \cap (B^c \cap C)$.

After working it out on a number line, I got: $\{4, 5, 6\}$. As it stands, the expression contains the integers that do not belong to the set $B$ that cross into $C$. This would result in $4, 5, 6$. ...
1
vote
1answer
26 views

Define the image of the function $f :\Bbb Z \times \Bbb N →\Bbb R$ given by $f(a, b) = \frac{a−4}{7b}$?

$\Bbb Z$ - integers $\Bbb N$ - natural numbers (starting from 1) $\Bbb R$ - real numbers I believe the answer is the set of real numbers ($\Bbb R$), seeing as $b$ will not equal $0$ as the set of ...
0
votes
2answers
38 views

Verifying a Proof for Spivak's Calculus Question (Chapter 2 Problem 9)

It says "Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then A contains all natural numbers $\ge n_0$". Am I allowed to construct another set ...
3
votes
1answer
44 views

Convergence of the integral $\int_0^1 \frac {1}{x\sqrt {1+x^\beta}}dx$

Is my integral-convergence contradiction proof valid? I have to brush up on my proof making. I am a little rusty. I was not sure if the following really held up. I wanted to prove the following is ...
2
votes
2answers
70 views

prove if f(x) has an infinite limit then limit of 1/f(x) is = 0

I wanted to ask if someone can do me the favor pointing out the mistakes I might of made in proving the theorem below. Also is there a way to prove the theorem without using the definition of limits? ...
1
vote
2answers
64 views

Proving $\frac{1}{4n}$ converges to $0$ in epsilon-delta form.

$$(\forall\varepsilon>0)(\exists K)(\forall x)(x>K\Rightarrow\frac{1}{4x}<\varepsilon)$$ Here is my attempted proof: Let $\varepsilon>0$ be a real number. Let $K$ be ...
1
vote
1answer
33 views

Divisibility proofs for greatest common divisor

I am studying divisibility and greatest common divisors. I have reached a section where I need to prove properties. My question is: are my proofs substantial? Or do I need to add to them? Below are ...
10
votes
1answer
173 views

Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-

Let $n$ be a positve integer. Prove that$$\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$$ for each non-negative ...
-1
votes
2answers
21 views

Confusion with the reconstruction conjecture?

After reading about the reconstruction conjecture for graphs, I came up with what I thought was a proof by contradiction. Consider the class $T$ of (isomorphism classes of) finite graphs, and the ...
0
votes
1answer
60 views

Mean value theorem for vector valued function (not integral form)

Let $f:U\to\mathbb R^m$ be differentiable with $U\subseteq \mathbb R^n$ being open and convex. If $f$ is absolutely continuous, then by fundamental theorem of calculus we have following version of ...
0
votes
0answers
33 views

Proof Verification: Show $\lim_{x\rightarrow\infty}f(x)=0$ given $f\in L^1$ and $f$ absolutely continuous.

so I am working through a packet of old exams and have came across three that are very similar. I was able to produce a proof (possibly wrong?) that works for all three, but it doesn't use all their ...
3
votes
0answers
64 views

Find two homeomorphic topological spaces and a bijective continuous map between them which is not homeomorphism.

I'm aware that it is duplicate, but I'd like to know whether my example is appropriate or not. Let our function $f$ be on the set $\mathbb{Q}\cap\mathbb{Z}$ induced by standard topology of a line. ...
0
votes
0answers
11 views

Solution of $\sigma(p^3q^2)=2\varphi(p^3q^2)$?

Please suggest me if I made any mistake in the following: It is given that $\sigma(p^2q^2)=2\varphi(p^2q^2)$ has no solution in positive integer if $p, q$ are distinct prime. Let us see if ...
0
votes
1answer
35 views

Induction proof question

Show by induction that for all integers n $\ge$ 1 $$ \sum_{i=1}^n i3^i = \frac{3(2n3^n-3^n+1)}{4} $$ Starting with n = 1 will give me LHS = 3 and RHS = 3. Inserting n = p gives $$\sum\limits_{i=1}^p ...
3
votes
0answers
45 views

Debunking an elementary proof of FLT

José Cayolla: Fermat's Last Theorem admits an infinity of proving ways and two corollaries. arXiv:1507.06989 [math.GM] I don't usually devote so much time to "crackpot papers", but I have a ...
3
votes
4answers
98 views

Prove that $n!>n^2$ for all integers $n \geq 4$.

I am working on induction problems to prep for Real Analysis for the fall semester. I wanted proof verification and editing suggestions for part (a), and assistance understanding part (b). For part ...
0
votes
1answer
34 views

Let $n$ and $p$ be positive integers. Show that $n$ can always be expressed in the form $n=pq+r$

I would have thought this would have been on here somewhere. Here I go. Let $S$ be the set of positive integers $n$ which can be expressed in the form $n = pq + r$ where $ 0 \leq r < p.$ where ...
3
votes
2answers
66 views

Is the following a correct logical proof?

A → (F ∧ P) ~A → (S ∧ R) ~R ∴ P     assume ~P         assume A         F ∧ P ...
2
votes
1answer
33 views

$C^1(\bar \Omega)$ is a Banach space

My professor gave a proof of the completeness of $(C^1(\bar \Omega),\|\cdot \|_{C^1})$ based on the fundamental theorem of calculus. I though about an alternative and I would like to know whether this ...
1
vote
1answer
74 views

Proof check $(\log\log n) /(\log n) $ approaches zero

Proof : If $|a| < 1$ then $(na^n)$ is a null sequence therefore if $b>1$ then ${n\over (b)^n}$ is a null sequence.There is always an $m$ such that for every $n > m$ $${n\over (b)^n} ...
4
votes
2answers
61 views

Prove: the countable product of regular topological spaces is regular.

Prove: the countable product of regular topological spaces is regular. Label the countable product of $X_i$ as $X$. Given $x \in X$ and $U$ a closed set s.t. $ x \notin U$, let's find disjoint ...
2
votes
1answer
59 views

Is the sequence $S_n=1+\sum_{m=2}^n \prod_{k=m}^n a_k$ bounded?

I have the following sequence $$S_n=1+\sum_{m=2}^n \prod_{k=m}^n a_k$$ where $\{a_k\}$ is a real positive sequence with the property that $$\lim_{n\to \infty}\left(\prod_{k=1}^n ...
0
votes
1answer
17 views

Determining cardinality and inverse

Let the function $\chi: P(Z) \to P(Z)$ be defined by $\chi(B) = B^c$ for any $B \in P(Z)$. (In other words, $\chi$ sends a subset $ B \subseteq Z$ to its complement, $B^c$, i.e. the set $Z - B$.) ...
2
votes
1answer
74 views

A group is abelian if and only if the center of the group is all the group

Isn't it the same to say that a group is abelian, and that the center of the group is all the group? I have an exercise to prove that this is true, and it's exactly one stroke for each direction of ...
0
votes
0answers
31 views

Is this true :${(a+ib)}^{(k+ij)}=0$ iff $0<a=k<1$ and $b<j$?

let $z=a+ib ,s=k+ij$ are two complex numbers and let $f(z,s)$ be a complex function defined as follow :$$f(z,s)=z^s={(a+ib)}^{(k+ij)}$$ and $a,b,j, k$ are non -nul real numbers . .After some ...
1
vote
0answers
33 views

Sifted colimits of models of a Lawvere theory.

While trying to prove that the monad associated to a Lawvere theory is finitary, I came across the following, which troubles me. Let $\mathcal A$ be small category and $\mathcal C$ be the full ...
1
vote
0answers
29 views

Why $\lim_{a\to\infty} \frac{ Q_0(a,b)}{ \sqrt{a e}}e^{\frac{(a-1)^2}{2}}\neq Q (b)$?

I’m trying to find a connection between Marcum-Q function, which is defined as: $$Q_M(a,b)=a^{1-M}e^{-\frac{a^2}{2}}\int_{b}^{\infty} x^M \exp^{-\frac{x^2}{2}} \mathrm I_{M-1}(a x)\mathrm dx$$ where ...
1
vote
1answer
40 views

Is my proof for this fact correct?

The thing ought to be proven Let $a$ and $b$ be nonzero integers that are relatively prime, and let $c$ be an integer. Show that $ax+by=c$ has an integer solution. My postulated proof that ought ...
2
votes
2answers
76 views

Proving that $P\left ( \bigcup_{i=1}^{n}A_{i} \right )\leq \sum_{i=1}^{n}P(A_{i})$ by induction

Proposition 1: Let $A_{1},\dots, A_{n}$ be events in the probability space $(\Omega,\mathcal{F},P)$. Then $$P\left ( \bigcup_{i=1}^{n}A_{i} \right )\leq \sum_{i=1}^{n}P(A_{i}).$$ Let's start with a ...
1
vote
1answer
54 views

A proof that the operation of concatenation has an identity element

I'm relatively new to proofing and am wondering if this is an acceptable proof. The book for anyone who would like to reference it is "A Book of Abstract Algebra" by Charles Pinter. It is problem ...
1
vote
2answers
49 views

Proving that $B \cap ((A \cup B) \cap (B' \cap A')') = B$ using set algebra

Problem: Use set operation laws to prove the following set equality, and clearly indicate which law(s) you use in each step: $$B ∩ ((A ∪ B) ∩ (B' ∩ A')') = B.$$ Answer: \begin{align} B ∩ ((A ∪ ...
6
votes
2answers
112 views

How do I show that $ \sin x, \cos x$ really are in $ [-1,1]$ using series notion?

I'm sorry to ask this question , but should ask it may help me to know more about series theory , It is well known that $\cos x $ and $\sin x$ are represented by alternative series which hard ...
1
vote
1answer
28 views

Few questions about the basics of Cardinality

I am looking for some help to either conform that my reasoning is sound, or to please elaborate to me more on the subject so I can gain a better understanding. I am studying some from my class notes, ...
1
vote
1answer
28 views

Analysis $\epsilon-N$ notation, switching between rigorous proof and intuition.

I once saw a proof for convergence for a specific problem involving fixed point iteration showing $|a_n-L| \leq (1/2)^n \rightarrow 0 $ and so $a_n\rightarrow L$ I initially did not understand why ...
1
vote
1answer
36 views

Finding standrad deviation $\sigma$

Carton of milk can be reserved fresh for $20$ days in average, $\frac13$ from the milk cartons can reserved fresh for $22$ days or more. Let's assume that the period of fresh is exponential ...
1
vote
1answer
75 views

Can you define the radius of convergence of a power series by an upper bound on the sequence of coefficients?

Let $P(z) = \sum_{n = 0}^\infty c_n z^n$ be a complex power series. Consider the follwing subsets of $\mathbb{R}$ $$ \begin{align} A_1 &:= \{r \geq 0 \,:\, (c_n r^n)_{n \in \mathbb{N}_0} \text{ ...
0
votes
0answers
37 views

minimum number of leaves in a perfect binary tree

I'm trying to prove that the number of leaves in a perfect binary tree is at least H+1 where H is the height of the tree. This is what I've done up til now: No of leaves at height $H = 2^H$ Base ...
4
votes
4answers
144 views

Proof of the Binomial theorem; does ${n-1}\choose{n}$ make sense?

I wanted to read a proof for the Binomial theorem, so I googled "proof of the binomial theorem". My question is about the proof from the top link of that search. In the sixth line of the induction ...
1
vote
2answers
31 views

Show that function is in L^2

I'm going through a paper and I came across the following statement: Given $\mathbf{q}_h \in \mathbf{V}_h(\Omega)$ we have to show that $\nabla\cdot\mathbf{q}_h$ is well defined and in ...
2
votes
3answers
82 views

How do I show this is a surjection?

Problem: Assume $f: \mathbb{N}_0 \rightarrow X$ and $g: \mathbb{N}_0 \rightarrow Y$ are bijections. Prove that the function $h: \mathbb{N}_0 \rightarrow X \cup Y$ defined as \begin{align*} h(n) = ...
2
votes
1answer
51 views

Proof of associativity for concatenation operation

I recently took a mathematical proof class and am beginning to teach myself abstract algebra. I'm fairly new to proofing however, and am not very confident in how I do it. Also, I'm new to this site ...