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Questions tagged [proof-verification]

For questions concerning a specific proof or a specific solution, asking for verification, identifying errors, suggestions for improvement, etc. (You should not use this tag if the question does not contain your proposed proof/solution.)

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1answer
9 views

Proof Verification: A monotonically increasing sequence that is bounded above always has a LUB

Problem: Prove that a monotonically increasing sequence that is bounded above always has a least upper bound. This theorem is present in calculus and real analysis books along with proofs for it. I ...
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0answers
5 views

Natural Deduction Logic

~(~U•~W) U>(X•E) (U>~E)•(W>~X) / ~U
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0answers
6 views

Variance of Product of Ind. Variables

Whats wrong with my approach to answer the following question? The number of customers arriving to a fast food restaurant between 7 am and 9 am has the Poisson distribution with mean 40. Suppose that ...
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0answers
23 views

Proving $¬∃x (P(x)) → ∀x (¬P(x))$

Just curious if my proof is correct: ...
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2answers
38 views

Fundamental Theorem of Calculus improper integral question

I want to find the derivative of a function G(x) defined by: $$ G(t) = \int_{0}^{t^2} x^{2} \sin(x^{2}) \, dx$$ Am I correct in evaluating the answer to be $= 2t^{3}\sin(t^{2})$? What I did was ...
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0answers
11 views

Proving a theorem about the transient of a function and the average value

Well, I want to prove the following theorem: Theorem: If the function $f(t)$ is the sum of two functions $y(t)$ and $z(t)$. And $y(t)$ is the transient part of the function $f(t)$, the average ...
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1answer
24 views

Showing a 1-form on the sphere $S^2$ can not be obtained from exterior derivative

I have the following problem: Can the vector field $X(x,y,z)=(-y,x,0)$ on $S^2$ be the gradient (on the sphere) of a function $f:S^2\rightarrow \mathbb{R}$ with respect to the standard euclidean ...
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1answer
30 views

Show that $g$ only have removable singularities in $\mathbb{D}$

I've been asked to show that $g(z)=\frac{\varphi_b (f(z))}{h(z)}$ only have removable singularities in $\mathbb{D}$, where $f: \mathbb{D} \rightarrow \mathbb{D}$ holomorphic with $f(0)=0$, $h(z) = \...
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1answer
33 views

Completeness of a specific Hilbert space

I am reading Zehnder's book "lectures on dynamical systems". In the chapter 7 he defines a space and states that it is a Hilbert space. I am struggling to show that the space is indeed complete. Here'...
2
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1answer
43 views

Trisecting an angle $\theta$ equally via applying trigonometry

I found an article in a book about trisecting an angle equally. It was written there that Archimedes tried to solve that process by applying pure geometry (using only compass and scale without its ...
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1answer
37 views

Proof verification there exist at most one $c\in[0,1]$ such that $f(c)=c$.

Let the function $f:[0,1]\rightarrow \mathbb{R}$ be continuous on $[0,1]$ and differentiable on $(0,1)$ and $|f'(x)|<1$ for $ \forall x \in (0,1)$ I want to prove this statement: there exist at ...
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0answers
66 views

A single set of moves $S$ that, if repeated, solves the Rubik's cube from any state

I am looking for a proof verification. I often find these concepts simple, but struggle to communicate them clearly. Communication in mathematics is very important to me: Examples could be: Any ...
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2answers
19 views

Show that $I=(−\infty,\sup I]$ as $I$ is bounded above but not bounded below

Let $I$ be a non-empty interval. Suppose $I$ is not bounded below, I is bounded above, and $\sup I ∈ I$. Show that $I=(−\infty,c]$, where $c=\sup I$. My attempt:($\Longrightarrow$) Since $I$ is not ...
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1answer
26 views

Proving a limit converges to -3 using definition of convergence.

so I have the problem $\lim\limits_{n \to oo} \frac{2-3n^2}{n^2+2n+1}$. I have to prove this using the epsilon definition. So I know the limit equals -3. So I do |$\frac{2-3n^2}{n^2+2n+1}$ + 3 | < $...
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1answer
22 views

Can partial derivatives be expressed as a fraction?

Let $a = bc$. Then $b = a/c$. From the first equation, we also have $\frac{\partial a}{\partial c} = b$. Equating, $\frac{\partial a}{\partial c} = b = a/c$, or $\frac{\partial a}{\partial c} = a/c$, ...
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2answers
33 views

Proving that $y_n$ converges given $y_{2n}, y_{2n+1}$ converges

Suppose we have $\mathrm{lim}_{n \rightarrow \infty} y_{2n} = \mathrm{lim}_{n \rightarrow \infty} y_{2n+1} = M \in \mathbb R$. I'm trying to prove from this that $\mathrm{lim}_{n \rightarrow \infty} ...
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2answers
40 views

Is this proof that $\text{diam}(A) = \text{diam}(\bar{A})$ correct?

Let $M$ be a metric space. I'm asked to prove that the diameter of a set $A\subset M$ is the same diameter as its closure $\bar{A}\subset M,$ $$\text{diam}(A) = \text{diam}(\bar{A}).$$ My attempt: ...
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4answers
57 views

Is my proof that $\overline{A}\cup\overline{B} = \overline{A\cup B}$ correct?

Let $\overline{A}$ define the closure of $A$. I'm asked to prove that $$\overline{A}\cup\overline{B} = \overline{A \cup B}.$$ My attempt at this: $\overline{A}\cup\overline{B}$ is the union of the ...
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1answer
24 views

Find $a_{n+2}+3a_{n+1}+2a_n=3^n$ if $a_0=0$ and $a_1=1$, and prove

Find $a_{n+2}+3a_{n+1}+2a_n=3^n$ if $a_0=0$ and $a_1=1$, and prove the result. What I have done: First we have to find the homogeneous recurrence relation solution, so $$a_{n+2}+3a_{n+1}+2a_n=0\...
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2answers
32 views

Integrate $\int_{|z|=1}z^{3}e^{1/z}dz$ - verification

I integrate over a circular path centered at 0 with radius 1 $\int_{|z|=1}z^{3}e^{1/z}dz=\int_{|z|=1}z^{3}\sum\limits_{n=0}^{\infty}\frac{1}{n!z^{n}}dz=\int_{|z|=1}\sum\limits_{n=0}^{\infty}\frac{1}{...
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2answers
44 views

If $D$ be the differentiation operator on $V$. Find $D^*$.

Let $V$ be the vector space of the polynomials over $R$ of degree less than or equal to $3$ with the inner product space $(f|g)=\int_{0} ^{1}f(t)g(t) dt$, and let $D$ be the differentiation operator ...
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3answers
29 views

Integrate $\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz$ verification

I integrate over the edge of a circle $K$ with radius 1/2 $\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz=\int_{|z|=1/2}-\frac{e^{1-z}}{z^{3}}\frac{1}{(z-1)}dz$ By the Cauchy Integral form $f(w)=\frac{...
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1answer
32 views

How many different subsets $Y \subset X$ with $|Y| \leq n$ are there if $|X| = 2n+1$? (Elegant solution?)

I've been given the above task, and I was quite surprised about how easy it seemed. My solution $$N = \binom{2n+1}{0} + \binom{2n+1}{1} + \ldots + \binom{2n+1}{n} = \sum_{i = 0}^{n} \binom{2n+1}{i}$$...
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0answers
28 views

Show that $T$ has an adjoint, and describe $T^*$ explicitly.

Let $V$ be an inner product space and $ \beta, \gamma$ fixed vectors in $V$. Show that $T \alpha = (\alpha\mid\beta) \gamma$ defines a linear operator on $V$. Show that $T$ has an adjoint, and ...
1
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1answer
21 views

What is the probability if we throw dart towards a large square but it should hit only the inner part of small square $FEHG$ inscribed in it?

Let $ABCD$ be a square shaped board. 4 equal rectangles are drawn into it. The length of the sides of the rectangles are $x$ and $y$, where $\frac{x}{y}$ = $3$. A dart is thrown towards the square ...
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1answer
24 views

Is the integral of a smooth function on a closed smooth real manifold finite?

Let $M$ be a closed smooth real manifold and $f:M\to M$ a $C^\infty(M)$ a function. I need to prove whether $$\int_M f(p)\mathbb dp\in\Bbb R$$ I think yes, because for all $p\in M$, $f(p)\leq k$ for ...
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1answer
33 views

Prove the equivalent conditions for nowhere dense subset.

Let $(X,d)$ be a metric space and $A$ be a subset of $X$. Then the following statements are equivalent. $A$ is nowhere dense. $\overline{A}$ doesn't contain any non-empty open set. Each ...
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3answers
32 views

Proof that if $A \subseteq B \setminus C$ then $A$ and $C$ are disjoint.

Here is my attempt at proving the theorem: Proof. Suppose $A \subseteq B \setminus C$. Let $x$ be an arbitrary element of $A$. We can conclude that $x \in B \setminus C$, since $A$ is a subset of $B \...
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0answers
46 views

Proof that continuous curve in $\mathbb{R}^2$ has Lebesgue measure zero

Suppose $\Gamma$ is a curve $y = f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0.$ [Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$.] My attempt: ...
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0answers
20 views

Every uncountable closed set contains a perfect subset

Every uncountable closed set contains a perfect subset. Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! My attempt: For $A \...
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1answer
17 views

Solving a 1D heat transfer ODE BVP problem $\frac{d}{dx}(k\frac{du}{dx})$ over three homogeneous layers.

Question I need help on part a of this question Context This is from a book on the finite element method. Finite elements a gentle introduction Chapter 4 question 5 From the answer at ...
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0answers
18 views

Find $T^* $, where $T$ is the linear operator defined by $T \epsilon_1 = (1, - 2), \,\,T\epsilon_2 =(i, - 1)$.

Let $V$ be the space $\mathbb C^2$, with the standard inner product. Let $T$ be the linear operator defined by $T \epsilon_1 = (1, - 2), \,\,\ T\epsilon_2 = (i, - 1)$. If $ \alpha = (x_1, x_2)$, find ...
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0answers
44 views

Solution verification: Writing a sum in the Cantor normal form

Can someone please help with the following problem. I need to write the following sum in Cantor normal form: $$\sum_{i ∈ ω\cdot2} \sum_{j ∈ i} (i+j)$$ The result I'm getting is $$ω^2 + w,$$ so I ...
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0answers
39 views

No continuous function $f$ on $\mathbb{R}$ such that $f(x)=1_{[0,1]}(x)$ almost everywhere

Let $1_{[0,1]}$ be the characteristic function of $[0,1]$. Show that there is no everywhere continuous function $f$ on $\mathbb{R}$ such that $$f(x)=1_{[0,1]}(x)\,\,\,\,\,\,\,\,\,\,\,\,\,\text{...
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1answer
33 views

Let ST be a chord of a circle ω which is not a diameter, and let A be a fixed point on ST

Let segment ST be a chord of a circle ω which is not a diameter, and let A be a fixed point on ST. For which point X on minor arc ST is the length AX minimized? What I did: I thought that the length ...
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2answers
32 views

Eigenvalue of a matrix $A$

WTS: A scalar $\lambda$ is an eigenvalue of a matrix $A$ $\iff$ $\det(\lambda I-A)=0$ My proof: Assume $\lambda$ is an eigenvalue of A. So $Av=\lambda v$ for a non-zero vector, v.This is equivalent ...
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0answers
8 views

Decomposition of subrepresentation

Let $K$ be a compact (Lie) group and let $(\pi,\mathcal H)$ be a $K$-representation such that the $K$-finite vectors $\mathcal H_K$ decompose as $\mathcal H_K=\bigoplus_{\gamma\in S\subseteq \widehat ...
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0answers
30 views

Verify that $f(x)=\frac{1}{x}-\frac{1}{x_0}$ is continuous for every $x_0\neq 0$.

Verify that $f(x)=\frac{1}{x}-\frac{1}{x_0}$ is continuous for every $x_0\neq 0$. $f(0)$ is not defined. So the function is discontinuous at $0$. Let $c\in \mathbb{R}\setminus \lbrace 0 \rbrace$, we ...
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0answers
34 views

An inequality about borel measures in a proof

I'm having trouble with the inequality in the proof of the following claim: Let $E$ be a locally-compact separable metric space and let $\{\mu_n\}$ weak-* converge to $\mu$. If $\{|\mu_n|\}$ ...
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0answers
21 views

Show that for every real number x, there exists an integer n so that $\ \left|n\ -x\right|\ \le\ \frac{1}{2}$

I am very new to proofs and just wanted to make sure that mine isn't absolute nonsense. Here's the question and what I wrote: Show that for every real number x, there exists an integer n so that $...
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2answers
28 views

A question on the equality between two sets.

Let $\{f_n\}_{n\in\mathbb{N}}$ a sequence of function on $X$ to $[-\infty,\infty]$ and let $\alpha\in\mathbb{R}$. I must prove that $$ \bigg\{x\in X\;\bigg|\sup_nf_n(x)>\alpha\bigg\}=\bigcup_{n=...
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1answer
48 views

Is a set of measure zero in $\mathbb{R}$ totally disconnected?

Let $M \subset \mathbb{R}$ be a nonempty set of Lebesgue measure zero. Does it follow that $M$ is totally disconnected in the sense that for any $x<y$, with $x,y\in M,$ there exists $z\notin M$ ...
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0answers
40 views

$\int_0^{+\infty} f(x) dx = - \lim_{a\rightarrow 0^+} \frac{1}{a} \sum Res_{\Bbb{C} \setminus \Bbb{R}^+} (f(z)z^a)$?

I find integrals on the real axes computed by complex number techniques and looking for a generalization working just on the positive semiaxes. I tried to put together this general result, but I am ...
5
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1answer
79 views

Have I proved in ZF $\aleph (X) \lt \aleph (\mathscr P^3(X))$?

As a reminder, I've been a lawyer for more than 25 years after bailing on graduate school. I recently reacquired the itch to do math. To that end, I've been (slowly) working through Set Theory: An ...
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0answers
46 views

Proof verification using Schwarz Lemma

Let $f : \mathbb{D} \rightarrow \mathbb{D}$ be an holomorphic function in $\mathbb{D}$ with $f(0)=0$. Let $a_1, a_2,\ldots,a_n$ be $n$ different points in $\mathbb{D}$ with $f(a_j)=b \; \forall \; j =...
1
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1answer
19 views

Find an orthonormal basis in which $E$ is represented by matrix

Let $W$ be the subspace of $R^2$ spanned by the vector $(3, 4)$. Using the standard inner product, let $E$ be the orthogonal projection of $R^2$ onto $W$. Find (a) a formula for $E(x_1, x_2)$ ; (b)...
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4answers
167 views

Find the area of the shaded region of two circle with the radius of $r_1$ and $r_2$

In the given figure , $O$ is the center of the circle and $r_1 =7cm$,$r_2=14cm,$ $\angle AOC =40^{\circ}$. Find the area of the shaded region My attempt: Area of shaded region $=\pi r^2_2 - \pi ...
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0answers
21 views

Proof check: Showing gcd(a,b) has prime factorization with exponents taken to be minimum of those for a,b

Using the notation from the previous problem in the text, we've that $$a=t_1^{g_1}t_2^{g_2}\ldots t_v^{g_v} \quad \text { and } \quad b = t_1^{h_1}t_2^{h_2}\ldots t_v^{h_v},$$ for $t_1 < t_2 < \...
1
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1answer
35 views

Find $3$ non-isomorphic groups of order $2012$

Find $3$ non-isomorphic groups of order $2012$. Is the following correct? First of all, we have the two non-isomorphic abelian groups $\mathbb Z_{2012}$ and $\mathbb Z_{2}\times\mathbb Z_{1006}$. ...
2
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0answers
30 views

Autonomous system of ODEs: Using Picard-Lindelöf after transforming to multiple non-autonomous ODEs

Assume I have an N-dimensional system of first-order autonomous ODEs $y'(t)=F(y(t))$ with $F:\mathbb{R}^N\rightarrow\mathbb{R}^N$ given as $F(y)=\begin{pmatrix}f_1(y_1,y_2,\ldots,y_N)\\f_2(y_1,y_2,...