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

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2
votes
1answer
50 views

Characteristic of a Ring not making sense.

The characteristic of a ring with unity is defined to be the least positive integer $n$ such that $1$ plus itself $n$ times $=0$. How does this make sense? $1$ plus itself $n$ times $=n1=n=0$, but ...
4
votes
2answers
270 views

Determinig whether a complicated series has a limit - is there any other way to do it?

Here's a series the limit of which I want to evaluate: $$a_n=\sum_{n=1}^{\infty}\frac{(1+\frac{1}{n})^nn^2-7n}{n^3+3n^2+1}$$ Here's how I did it. Let's define 3 sequences: ...
0
votes
1answer
21 views

Prove uniform convergence if continuous and converge uniformly on a dense set

If $A \subset E \subset \mathbb{R}$, then $A$ is dense in $E$ in case $E = \overline{A} \bigcap E$. Assume$\{f_n\}_{n=1}^\infty$ is a sequence of functions continuous on $E$ and converging uniformly ...
0
votes
0answers
31 views

Show that $\displaystyle\sum_{i=0}^r\frac{\gamma_i^{\star}}{r+1-i}=0$

Let $\displaystyle\gamma_i=\int_0^1\frac{s(s+1)...(s+i-1)}{i!}ds$ and $\displaystyle\gamma_i^{\star}=\int_{-1}^0\frac{s(s+1)...(s+i-1)}{i!}ds$ If ...
1
vote
1answer
59 views

Prove that the product space is a metric space.

I have the following problem: Let $(S,d)$ and $(T,e)$ be two metric spaces. Their product space has underlying set $$S\times T=\{(s,t)|s\in S,t\in T\}$$ and metric ...
1
vote
0answers
9 views

Verification of proof that the left distributive property holds on a field of quotients, F

Here's my proof. I would like to know whether it is right or wrong and if the latter where it needs to be corrected: (a,b)=[(a,b)][(cf+ed,df)]=[(acf+aed,bdf)]. We want to show its equivalence to: ...
0
votes
0answers
40 views

How can I prove this theorem?

Let n ∈ N. Let b ∈ Z. Then there exists c ∈ Z satisfying c ·n b = 1
1
vote
1answer
16 views

Identity Tranformation Proof- Is this enough to prove this statement?

Let {v$_1$,...,v$_n$} be a basis for a vector space V and let T:V$\to$V be a linear transformation. Prove that if T(v$_1$)= v$_1$,...,T(v$_n$)= v$_n$, then T is the identity tranformation on V. I'm ...
0
votes
0answers
30 views

Check: Find all the non-isomorphic Abelian groups of order $200$.

Find all the non-isomorphic Abelian groups of order $200$. Since $200=5^2 \cdot 2^3$, there are $3 \times 2=6$ such groups. The six partitions are: $25 \times 8$ $25 \times (4 \times 2)$ $25 \times ...
2
votes
2answers
72 views

If $p_n\to p$ in a metric space, then the set of points $\{p,p_1, p_2, p_3, \ldots \}$ is closed

I'm having trouble with limits and convergence, bear with me. Prove that if $\lim_{n \to +\infty} p_n = p$ in a given metric space, then the set of points $\{p,p_1, p_2, p_3, \ldots \}$ is ...
1
vote
1answer
42 views

Open and Closed Sets in $\mathbb E^2$

I'm having problems trying to prove open and closed sets so here goes: Prove that $\{(x, y) \in \mathbb E^2 : x > y\}$ is an open set Prove that $\{(x,y) \in \mathbb E^2 : xy = 1 \}$ is a closed ...
0
votes
0answers
10 views

Is the following the idea behind the antisymetric property of the order on ordanals

I am trying to understand the claim that the anti-symmetry property of the order on ordinals is a consequence of the fact that no well ordered set is order isomorphic to one of its segments. Could ...
1
vote
2answers
30 views

Simple Combinatorial Proof [duplicate]

$$ \sum_{i=0}^n {n \choose i} i = n2^{n-1} $$ I am having trouble formulating a combinatorial proof. An algebraic proof is quite simple, where one expands $(1 + x)^{n}$ and then takes the ...
5
votes
1answer
72 views

Question regarding Radon-Nikodym derivative…

The problems are as follows: (1) Let $X=[0,1]$ with Lebesuge measure and consider probability measures $\nu,\mu$ given by densities $f,g$ as follows: $$\nu(E)=\int_{E} ...
3
votes
3answers
72 views

Show that $\int_{a}^{c}f(x)dx = 0$ for all $c\in [a,b]$ if and only if $f(x) = 0$ for all $x\in [a,b]$.

Exercise: Suppose that $a<b$ and that $f:[a,b]\rightarrow R$ is continuous. Show that $\int_{a}^{c}f(x)dx = 0$ for all $c\in [a,b]$ if and only if $f(x) = 0$ for all $x\in [a,b]$. attempt of ...
1
vote
3answers
40 views

How do I prove that $\lim _{n\to \infty} (\sqrt{n^2 + 1} - n) = 0$?

As you can see from the title, I need help proving that $\lim _{n\to \infty} (\sqrt{n^2 + 1} - n) = 0$. I first looked for $N$ by using $|\sqrt{n^2 + 1} - n - 0| = \sqrt{n^2 + 1} - n < \varepsilon$ ...
1
vote
3answers
45 views

If $f: A→B$ and $g: B→C$ are surjective, then $g\circ f$ is surjective.

In my homework, I wrote: Assume f and g are surjective. Let m be an element of C. then there exists a b that's an element of B, such that g(b) = m and an a element of A such that f(a) = b by ...
0
votes
1answer
31 views

Suppose a, b, n ∈ N. Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b)

I had this for homework before, and I wrote: There exists s,t, that are elements of integers, such that gcd(a,b) = sa + tb, so ngcd(a,b) = n(sa + tb) = s(na) + t(nb) but I got it wrong. How do I ...
0
votes
1answer
47 views

Proof by induction of this formula? [duplicate]

$2^0+2^1+2^2+...+2^n$ for $n ∈ \mathbb{N}$ U ${0}$. I made a conjecture that this is $2^{n+1} - 1$. Now I have to prove it by induction. I tested the base case where it's equal to zero, and it ...
0
votes
1answer
36 views

$\int_{J}(f(x))^{2} dx = \int_{J}(g(x))^{2} dx \Rightarrow \int_{J}|f(x)| dx = \int_{J}|g(x)|$

Let $f,g$ be smooth functions. Then the following applies:$$\int_{J}(f(x))^{2} dx = \int_{J}(g(x))^{2} dx \Rightarrow \int_{J}|f(x)| dx = \int_{J}|g(x)|$$ for $J$ interval. Here is the proof: ...
1
vote
0answers
21 views

Gradient points in the direction of greatest change

Can anyone provide me with an alternative, possibly more intuitive proof of this proposition? I'm confused with where $cos\theta$ has come from?
3
votes
4answers
94 views

Proof check for Putnam practice problem

I realize this is simply an A1 problem, but my proof seems way too simple, so I would like someone to point out whether or not it's correct (and most importantly, fix any flaws in it). Problem. ...
0
votes
2answers
24 views

Trouble understanding surjective function proof

I'm studying for my discrete math exam and I'm having some trouble understanding this practice problem and solution. I know what surjective functions are, but I can't really understand the way this ...
0
votes
1answer
26 views

Verification of proof that for a,b in ring R, assuming ab is a zero divisor at least one of a and b is zero divisor

I'm not so sure if this is correct but here's what I have so far: ab is a zero divisor iff there is a c$\neq$0 s.t. (ab)c=c(ab)=0 given ab$\neq$0 and c$\neq$0. Then we have ...
3
votes
1answer
49 views

Proof Verification of Schröder–Bernstein theorem

So I've spent some time studying the Schröder–Bernstein theorem, but I'm trying to do the exercise in "Naive Set Theory" by Paul Halmos regarding the theorem. The exercise is finding an alternative ...
0
votes
2answers
49 views

Verification of proof that if $R$ is a commutative ring, $a$ is a unit and $b^2=0$ then $a+b$ is also a unit

Here's what I have so far and I would like to know if I am right or if my proof needs to be edited: Since $a$ is a unit it means $a1=a$, with $1$ being the unity element We know $b^2=0$ and this ...
1
vote
1answer
36 views

Verification of Proof: Let $R$ be a ring of unity and $a \in R$ satisfy $a^2=1$. $S=\{ara \mid r \in R\}$ is a subring

Here's what I got. The three conditions we have to prove are: $0$ is in $S$: Let $r=0$ and this implies $a0a=0a=0$ which is in $S$ $(a-b)$ is in $S$ for all $a,b \in S$: Let $a, b \in S$. this ...
1
vote
1answer
20 views

Proof of continuity of stochastic processes defined by Ito integrals

I'm currently trying to understand the proof of Theorem 4.6.2 in Kuo, Hui-Hsiung: Introduction to Stochastic Integration: Suppose $f \in L^2_{ad} ([a,b] \times \Omega )$, then the stochastic ...
2
votes
1answer
31 views

Problem about supremum and upper bounds

Let $ \mathbb{R} \supset E \neq \varnothing$. Let $u \in \mathbb{R}$. Suppose $1.$ for every $n \in \mathbb{N}$, $u - \frac{1}{n} $ is not an upper bound of $E$. $2.$ for every $n \in \mathbb{N}$, ...
2
votes
0answers
33 views

Proofchecking: Application of Banach-Alaoglu on weak converging nullsequence

Problem Assume $x_n \to 0$ weakly in a Banach space. Show that for all $\epsilon>0$ and for all $N\in \mathbb{N}$ there exists a $n>N$ s.t. for all $f\in X^\ast, \|f\|\leq 1$ there exists ...
1
vote
0answers
41 views

Damped pendulum, Lyapunov function

The question is about the damped pendulum, There are $2$ statements, I don't understand or I'm not sure if my justification for them is correct, can you say please if I'm right, the example is from a ...
0
votes
1answer
27 views

A question about an exercise on basic cardinal arithmetic.

I just want to make sure that I have proved the following exercise correctly. Given two cardinal numbers $a$ and $b$ where $a$ is infinite. I was to show that $2\le b \le 2^a \implies b^a=2^a$ I ...
4
votes
2answers
38 views

Prove by induction that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{n+2}{2^n}$

Prove by induction that $$ \frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{n+2}{2^n} $$ Let $n=r$, so that $$ S_r=2-\frac{r+2}{2^r} $$ Therefore $$\begin{align} ...
0
votes
1answer
43 views

Is this proof about clock hands lining up correct?

Is http://joshuaoldenburg.com/articles/clock-hands-line-up/ a proof? I.e. does it sufficiently prove the times where the clock hands line up? $$ \begin{align} H &= \text{hour (1-12)} \\ M &= ...
3
votes
1answer
65 views

Prove that any subfield of $\mathbb C$ must contain $\mathbb Q$

I just started reading Linear Algebra by Hoffman and Kunze, and I came across the following line: The interested reader should verify that any subfield of $\mathbb C$ must contain every ...
0
votes
1answer
26 views

Adding Sequences proof

Prove that if $a,b : \mathbb{N} \to \mathbb{R} $ are sequences with $\lim_{n \to \infty}{a_n} = L$ and $\lim_{n \to \infty}{b_n} = M$ then $\lim_{n \to \infty}{a_n+b_n} = L + M$ Prove that if ...
1
vote
0answers
34 views

Can someone check my answer to a measure theory question on existence and equality of three integrals.

I have been told to investigate the existence and equality of the integrals; $\int_{[0,1]^2} f\;d\lambda^2$, $\int_0^1\int_0^1 f\;d\lambda(x)d\lambda(y)$ and $\int_0^1\int_0^1 ...
0
votes
1answer
27 views

$f$ is integrable on $[a,b]$ then $\int_{x}^{x+h}\frac{|f(t)-f(x)|}{h}dt\to 0$ if $h\to 0$ for almost all $x$

I am trying to prove if $f$ is integrable on $[a,b]$ then $\int_{x}^{x+h}\frac{|f(t)-f(x)|}{h}dt\to 0$ if $h\to 0$ for almost all $x$ but I am not sure. Attempt - Let ...
2
votes
2answers
36 views

Very easy quiz problem regarding the Archemedean property of R

I had this problem in my quiz on Friday: Consider the following statements: $1$. If $x \in \mathbb{R} $, then there exists some $n \in \mathbb{N}$ with $x < n $. $2.$ If $x,y \in \mathbb{R} $ ...
2
votes
0answers
75 views

Limit of $(\frac{1}{n^2})^{(\frac{1}{n})}$

Question: is this a valid procedure for finding the limit of $$\bigg(\frac{1}{n^2}\bigg)^{(\frac{1}{n})}$$ as $n\rightarrow \infty$. background: This is one of the examples of the failure of the ...
0
votes
1answer
42 views

Prove The Limit Does Not Exist

So I have a few questions in which I have to prove that the given sequence does not have a limit and I'm not too sure if I'm on the right track and if I am what is the next step that I have to do. Can ...
0
votes
0answers
32 views

Prove the limit exists

So I have a couple of problems in which I have to prove that the given limit exists and I'm not too sure if I'm on the right track and if I am what it is that I have to do next. Can anybody give me ...
0
votes
1answer
44 views

Convergent Series 2n-1/2n

Prove the series defined by P(n) = (1 *3 * 5 * (2n-1))/(2*4*6 * (2n)) is convergent It is monotone decreasing and bounded below by zero, but is that enough to say?
1
vote
2answers
60 views

Proving $\lim_{x\to \infty}\ln(x)/x$

Can you please check if my proofs are correct? for $$\lim_{x\to +\infty}\ln(x)=+\infty$$ I used the mean value theorem : $\ln$ continuous on $[1,x]$ $\ln$ differentiable on $(1,x)$ then there ...
0
votes
1answer
14 views

Verification of proof regarding limit and derivative at infinity

Ok so I have been working through Calculus by Spivak and stumbled upon a theorem which I found hard to prove ,and solution in answer book seems to be wrong.So I need you to help me verify my proof. ...
1
vote
2answers
29 views

Symmetric bilinear forms, quadratic forms and matrices

I have computed B=$ \left( \begin{array}{ccc} 0 & 4 & -1 \\ 4 & 2 & 3 \\ -1 & 3 & 1 \end{array} \right) $ Is this correct? If so, even though I may have achieved the correct ...
2
votes
1answer
34 views

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ smooth, $ g(x,y)= x^3 + y^3$ and $g \circ f \equiv 0$, then $\det Df \equiv 0$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function and $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $(x,y) \mapsto x^3 + y^3$. Assume that $g \circ f$ is identically $0$. ...
1
vote
1answer
30 views

A trivial question regarding the supremum of a set

Let $E$ be a subset of nonnegative real numbers. Suppose $E$ is bounded above and put $T = \{ x^2 : x \in E \}$. Say $\alpha = \sup E $. Then, $\alpha^2 = \sup T $. Attempt: Since $E$ is bounded ...
2
votes
0answers
20 views

A different characterization of the infimum of a set

Let $E$ be a set that is bounded below. Let $l$ be a lower bound of $E$. Show that $ l = \inf E $ iff given any $\epsilon > 0$ we can always find $z \in E$ with $z < l + \epsilon $. Attempt ...
1
vote
0answers
35 views

If $u\notin E$, then the supremum of $E\cup\{u\}$ is $\sup\{\sup E, u\}$

Let $S$ be an ordered set and $S \supset E $. Let $\alpha = \sup E \in E$. If $u \notin E$, then we have $ \sup ( E \cup \{u\} ) = \sup\{ \alpha, u \} $. Try: I know this result follows easy by ...