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

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30 views

What is this called specifically?

Imagine you take a radius from the center of the shape, you add up all of the lines as it rotates 360 degrees. The radius is measured from its point of rotation, like (0,0) in Cartesian coordinates,to ...
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1answer
24 views

3D Cauchy Problem

I will answer the question myself but let me know what you think of my correctness. We have the Cauchy Problem $$ yu_x-xu_y+u_z=0 $$ with data $u(x,y,0) = x+y$.
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0answers
21 views

Verify Result of a Calculation

In the journal: "A Closed Form Solution for the Similarity Transformation Parameters of Two Planar Point Sets", I cannot get same value for scaling factor for the same problem in the journal. Here is ...
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1answer
47 views

A question about the proof of an obvious result

This is obviously true that a local homeomorphism is a continuous map. I tried to prove it this way : Suppose $f:X \to Y$ is a local homeomorphism, then $f$ is continuous if for each $x\in X$ and ...
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0answers
36 views

Is my solution correct

I saw this question on a page: A kangaroo occupies an area of $50$ square centimetres and has a volume of $250000$ cubic centimetres. The kangaroo is $16$ metres away from a fence. Assume the ...
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2answers
61 views

How to prove indicator function, step function, and then for sequences of step functions?

I try to proof a claim. It should be done first for indicator functions, then for step functions and finally for limits of increasing sequences of step functions. I'm not sure if I'm doing it right. ...
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1answer
44 views

Eigenvectors for normal operators and their adjoints

Can someone tell me if this proof is correct? Claim:V is a vector space over the Complex field. $T:V\rightarrow V$ is a normal operator. Then if $v\in V$ is an eigenvector with the eigenvalue ...
4
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1answer
63 views

Is this a valid proof of $\lim _{n\rightarrow \infty }(1+\frac{z}{n})^n=e^z$?

Define the function $g_n(z)=(1+\frac{z}{n})^n$ for $\:n\in \mathbb{R^+}$. Then $\frac{d}{dz}g_n(z)=n(1+\frac{z}{n})^{n-1}\cdot\frac{1}{n}=(1+\frac{z}{n})^{n-1}$ Define $g_{\infty}(z)=\lim ...
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0answers
54 views

Find the flaw in my 1-page proof of the Four Color Theorem

The Four Color Theorem has been proven for quite a while now, so I'm not really breaking ground there. But last night, for some reason, it popped into my head and I started thinking about it. I feel I ...
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3answers
55 views

If $A,B$ are invertible so $AB$ is invertible

I thought about the following proofs but I am not sure about them there is $C,D$ so that $AC=CA=I$ and $BD=DB=I \rightarrow CABD=I \rightarrow$ due to associativity roles is no matrix $E$ so that ...
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2answers
61 views

Prove Or Disprove: tr(AB)=tr(A)*tr(B)

$\mathrm{tr}(AB)=\sum\limits_{i=1}^n \sum\limits_{j=1}^n a_{ij}*b_{ji}$ $\mathrm{tr}(A)*\mathrm{tr}(B)=\sum\limits_{i=1}^n a_{ii}*\sum\limits_{i=1}^n b_{ii}$ Therefore $\mathrm{tr}(AB) \neq ...
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3answers
81 views

If $x^n = f(x)g(x)$ as complex polynomials, must $f,g$ be of the form $x^m$?

If $x^n = f(x)g(x)$ as complex polynomials, must $f,g$ be of the form $x^m$? i.e. $x^n = x^mx^l$ where $m+l = n$. This is quite trivial, but I want to make sure I didn't miss anything. Attempt: If ...
1
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0answers
35 views

Passing of the limit for Lebesgue Integral (Proof Verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
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0answers
36 views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
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2answers
49 views

If $\lim_{x \to \infty}f'(x)=+\infty$ then $\lim_{x \to \infty}(f(x)-f(x-1))=+\infty$ and $\lim_{x \to \infty}f(x)=+\infty$.

Let $f$ be differentiable and let $\lim_{x \to \infty}f'(x)=+\infty$ prove that: 1) $\lim_{x \to \infty}(f(x)-f(x-1))=+\infty$ and 2) $\lim_{x \to \infty}f(x)=+\infty$. 1) I'll prove by ...
1
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1answer
32 views

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$.

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$. I need to verify correctness of my proof and ask if there is a more straight-forward ...
2
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0answers
77 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
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1answer
46 views

About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)

Problem (Rudin, R&CA chapter 2, no. 25) (i) Find the smallest positive constant $c$ such that $$ \log(1+e^t) \le c+t , \qquad t \in (0,+\infty). $$ (ii) Does $$ \lim_{n ...
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1answer
47 views

Need help with checking my proof related to equivalence mod n.

Let $n = a_ka_{k-1} \ldots a_1a_0$ be a natural number in base $10$. If $m = a_k + a_{k-1} + \ldots + a_1 + a_0$, then $n\equiv m\pmod n$. $Proof:$ Let $v = 0$, $x =a_k + a_{k-1} + \ldots + a_1$, ...
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1answer
72 views

Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
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3answers
447 views

Is this proof of the fundamental theorem of calculus correct?

A student friend of mine recently gave me a proof of the fundamental theorem of calculus which does not correspond to any I can find in the textbooks. It starts by considering an increasing continuous ...
4
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1answer
34 views

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$. Is $\cup \tau_\alpha$ a topology on $X$?

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$. Is $\cup \tau_\alpha$ a topology on $X$? For all $\alpha$, $\varnothing \in \tau_\alpha$ ...
4
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0answers
59 views

Two question about how to compute this integral limit

Let $f: (-\pi,\pi]\to \mathbb R$ be continuous and let $p_x (u) = {(f(u+x) - f(x)) \cos ({u \over 2}) \over \sin ({u \over 2}) }$. I want to show that $$ \int_{-\pi}^\pi p_x(u) \sin (Nu) du \to 0$$ ...
2
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1answer
23 views

Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?

Can someone please verify my proof? Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$? No. Let $X = \mathbb{R}$. Clearly, $\{x\} \in \tau_\infty$ ...
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2answers
36 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
2
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0answers
44 views

Critique of complex analysis proof

I'm working on the following complex analysis problem and am wondering if someone could critique my proof: Suppose that $f$ is an analytic function on some domain $D$ and that there exists a ...
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0answers
50 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
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1answer
30 views

Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
2
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1answer
79 views

Checking proof that $f(x)=x^2+1$ is continuous

Let $f:\mathbb R \to \mathbb R$ is defined by $f(x)=x^2+1$. Prove this function is continuous for all $x \in\mathbb R$. Here is what I have: Suppose that $c∈ℝ$. Let $\varepsilon>0$. Let ...
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1answer
36 views

Operator $Au(t) = \int_0^t e^{t-s} u(s) ds$ (Proof Verification)

Consider the space $C([0,1])$ with $||\cdot||_\infty$ norm. Let $A: C([0,1])\rightarrow C([0,1])$ be the operator defined by $$Au(t) = \int_0^t e^{t-s} u(s) ds.$$ And I am not 100% sure about (c), ...
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3answers
35 views

Considering $\epsilon$ intuitively in limit proof

I'm having rather difficult time in trying to use $\epsilon$ argument appropriately. For example here is my simple $\epsilon$ proof in one question. The question is as follow: Prove if $s_n \geq 0$ ...
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5answers
49 views

Proof of $\forall n \in \Bbb N$, $n > 2 \implies n! < n^n$

What I've got so far is this: Base case: n = 3 then $3 *2 * 1 = 6$ and $3^3 = 27$ $\therefore 6 < 27, 3! < 3^3$ So the base case is true. So if we assume $n! < n^n$ (n > 2) $(n + 1)! = ...
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3answers
102 views

Strange proof of Schwarz Inequality with Pythagorean Theorem

Does anyone know what is going on in this proof of the Schwarz inequality? Most importantly: how can one assume that $c^2\leqq \|A\|^2$, or later on, that $c^2\|B\| \leqq \|A\|^2$? This would imply ...
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1answer
76 views

Prove: $x^n=0 \to x=0$

I must prove the following: Prop. : let $x \in \Bbb{R}, n \in \Bbb{N}-\{0\}$ then $$x^n=0 \to x=0$$ Proof : by contradiction I have $x \neq 0$, by trichotomy one of the following holds $x <0 $ ...
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2answers
39 views

Proving Fibonacci inequality

I didn't see a question regarding this particular inequality, but I think that I have shown by induction that, for $n>1$. I am hoping someone can verify this proof. ...
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1answer
35 views

Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
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1answer
54 views

Is my proof on showing that the set of monotone functions on $[a,b]$ has cardinality of continum correct?

I was given an exercise problem to show that the cardinality of the set of all monotone functions on $[a,b]$ is $\aleph$. I came out with a proof which I am not sure if it is correct. My proof: Let ...
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0answers
16 views

Compactness of the projection operator

Let $H$ be a Hilbert space and let $F$ be a closed subspace. Then I'm to prove that the projection $p:H\rightarrow F$ is a compact operator if and only if $F$ is finite-dimensional. It's easy to prove ...
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3answers
105 views

My proof of the recursion principle (without the axiom of replacement)

(The proof in my book uses the axiom of replacement. My proof doesn't use it. Any hints and recommendations are welcomed.) The recursion principle Let $y_0$ be any element of a set $Y$ and ...
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0answers
24 views

Mobius function verification

I am looking to verify my answer to the question $$F(n)=\sum_{d|n}{\mu(d)\sigma(d)}=(-1)^{\omega(n)}\prod_{j=1}^{\omega(n)}{p_j}$$ Where $\mu$ is the Mobius function, $\sigma$ is the sum of divisors ...
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3answers
95 views

Discrete Mathematics Function Proof

The question is as follows : Let $f:A\rightarrow B$ be a surjective function and let $C$ be a subset of $B$. Prove $f(f^{-1}(C)) = C$. I understand what the question is asking. It's basically ...
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2answers
48 views

Discrete Mathmatics Proof

Here is the question: $a$ and $b$ are any two integers. $c$ is any prime. Prove that if $c$ divides $ab$, then $c$ divides $a$ or $c$ divides $b$ (or both, as in it can divide either or both, i.e. ...
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0answers
46 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
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5answers
371 views

Where's the problem with a false “proof”: $\;1^0 = 1^2 \overset{?}\implies 0 = 2$

What's wrong with this: $$\large 1^0=1^2$$ Since bases are same, therefore $$\large 0=2$$ My thinking: Since the function $\,f(x)=1^x\,$ is not one to one, therefore whenever $\,f(x)=f(y),\,$ ...
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1answer
24 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
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0answers
19 views

Orthonormality and fourier transform

If $g\in\mathcal{L}^2(\mathbb{R})$ then $\sum_{k\in\mathbb{Z}} |\hat{g}(\zeta+2k\pi)|^2=1$ for a.e $\zeta\in \mathbb{R} \Rightarrow \{g(.-k): k\in \mathbb{Z}\}$ is an orthonormal system. Please ...
2
votes
2answers
87 views

A problem on nested radicals

Find the value of $x$ for all $a>b^2$ if: $$\large x=\sqrt{a-b\sqrt{a+b\sqrt{a-b{\sqrt{a+b.......}}}}}$$ My attempt $$\large x=\sqrt{a-b\sqrt{(a+b)x}}$$ $$\large x^4=(a-b)^2(a+b)x$$ $$\large ...
2
votes
2answers
64 views

A problem on continued fractions

Find the value of $x$, if: $$\large 1+\frac{1}{2+\frac{1}{1+\frac{1}{2+...}}}$$ My attempt: Noting that: $$\large x=1+\frac{1}{2+\frac{1}{x}}$$ $$x=\frac{1+\sqrt{3}}{2}$$ question: Is my solution ...
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1answer
41 views

Proving some properties of $\Bbb N$ without using recursion

I will try to prove that if $a, b, c \in \Bbb N$ and $a \in b \in c$, then $a \in c$ (the transitivity property). I will not use recursion or the replacement axiom. Actually we can proove in the same ...
1
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2answers
87 views

Prove: If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

Can someone tell me if this proof is acceptable? Assume $A \not\subseteq C$. This means that there is an $x \in A$ such that $x \not\in C$. But since $\forall x \in A: x \in B$ and $\forall x \in B: ...