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

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3
votes
2answers
37 views

Construct $f: X\to Y$ such that $f(p)=p$

Let , $X=[-1,1]\times [-1,1]$ and $Y=\{0\}\times \left[-\frac{1}{2},\frac{1}{2}\right]$. Construct an example of a continuous map $f:X\to Y$ such that $f(p)=p$ for each $p\in Y$. I construct a ...
0
votes
0answers
15 views

Give an example of limits that misbehave under conjugation of function

My quest: Find real valued functions $f(x)$ and $g(x)$ such that $f \rightarrow b$ as $x\rightarrow a$ and $g\rightarrow c$ as $x\rightarrow b$ but $g(f(x)) \nrightarrow c$ as $x\rightarrow a$ I ...
0
votes
1answer
23 views

Strictly monotone functions and continuity

Let $f : X \to I$ be a strictly monotone surjective function mapping $X \subseteq \mathbb{R}$ to an interval $I \subseteq \mathbb{R}$. Then is $f$ necessarily continuous? Without loss of ...
0
votes
1answer
20 views

Real Analysis, Folland 3.4.26, Differentiation on Euclidean Space

Background Information - A Borel measure $\nu$ on $\mathbb{R}^n$ will be called regular if i.) $\nu(K) < \infty$ for every compact $K$ ii.) $\nu (E) = \inf\{\nu(U): E\subset U, U \ ...
2
votes
4answers
31 views

Showing that Harmonic numbers are $\Theta(\log n)$, intuitively

I wish to verify that Harmonic numbers $H_n = \sum_{k=1}^{n} \frac{1}{k}$ are $\Theta(\log n)$. One idea I have is to approximate the sum with an integral: $$\int_{1}^{n} \frac{1}{k} ~dk = \log(n) - ...
3
votes
0answers
26 views

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$).

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$). My effort Rearranging our equation we have : \begin{array}{c} 1990[x]+1989[-x]&=1 \\ ...
0
votes
2answers
27 views

Congruence problem $12x\equiv3\pmod{45}$ [on hold]

$$12x\equiv3\pmod{45}$$ Find all possible solutions to above congruence and show procedure in detail.
0
votes
0answers
32 views

Probability that a given function is prime…

If we have a set of primes $p_1$, $p_2$, ... , $p_n$, we can easily construct a function of their product: $$f(\alpha) = \alpha \left( \prod_{k=1}^n{p_k} \right) + 1, \alpha \in \mathbb{N}$$ I'm ...
0
votes
1answer
19 views

Limit of a sequence of measurable functions is measurable: alternative proof

If $f_n$ is an infinite sequence of measurable functions, let $\lim\limits_{n \rightarrow \infty} f_n(x) = f(x)$. Prove that $f$ is measurable. The proof (as I was taught it) is as follows. If ...
0
votes
1answer
56 views

Is there anything wrong in the following proof?

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces and let $U\subseteq X$ and $V\subseteq Y$ such that $U$ and $V$ are respectively open in $X$ and $Y$. Show that $U\times V$ is open in ...
-6
votes
0answers
32 views

Prove that $\sqrt 5$ is irrational given that Let $n$ be an integer. [on hold]

Prove that $\sqrt{5}$ is irrational given that Let $n$ be an integer. Prove that if $5|n^2$ then $5|n$. Hint: consider the contrapositve and consider cases. Also I considered the fact that $n$ cannot ...
0
votes
2answers
37 views

What is my mistake

Spot my mistake: $$\frac{\left(\text{P}_1+\text{P}_2+\dots+\text{P}_n\right)-\left(\text{Z}_1+\text{Z}_2+\dots+\text{Z}_n\right)}{n-m}\le-\ln(50)$$ ...
1
vote
1answer
42 views

Linear Alg. Short proof on determinant

Hi can I get a quick check on my proof to see if it is correct. proof
0
votes
2answers
39 views

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $\prod_{n=1}^\infty (1-a_n)$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$.

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $$\prod_{n=1}^\infty (1-a_n)$$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$. I know this question is similar to one I just ...
0
votes
4answers
74 views

Is this argument valid for a proof?

Please kindly forgive me if my question is too naive, i'm just a prospective undergraduate who is simply and deeply fascinated by the world of numbers. My question is: Suppose we want to prove that ...
2
votes
1answer
46 views

Deductive logic counter-intuitive result

I am working on a small proof in deductive logic. Here is what must be proved: $(\exists x \in T \mid A \implies P(x)) \implies A \implies (\forall x \in T \mid P(x))$ To me that looks unprovable ...
0
votes
1answer
12 views

Uniformly convergent in a set implies uniformly convergent in the set closure, too.

Let $f_n$:$X\rightarrow \mathbb{R}$ be a sequence of functions uniformly convergent in $X\subseteq \mathbb{R}$ . Suppose that each $f_n$ is continuous in the closure of $X$. Then $f_n$ is also ...
3
votes
1answer
39 views

Differential equation where Picard-Lindelöf can not be applied

My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function and let $u:[a,b]\to\mathbb{R}$ be a $C^1$ function such that $$\forall t\in[a,b],u'(t)=f(u(t))\text{ and ...
1
vote
1answer
48 views

Hartshorne Exercise II.2.18(d)

The Exercise: Let $\phi: A \rightarrow B$ be a ring homomorphism and let $X = \operatorname{Spec} A, Y = \operatorname{Spec} B$. Let $f: Y \rightarrow X$ be the morphism of schemes induced by $\phi$. ...
0
votes
1answer
40 views

Finding a closed formula for: $1\cdot2\cdot3+2\cdot3\cdot4+…+(n-2)\cdot(n-1)\cdot(n)$ [duplicate]

As I calculated the sum of the serie above doesn't exist(sum doesn't converge). How can I prove it using the double computing(combinatorical method)?
2
votes
3answers
64 views

Alternative Proof that $\sqrt{p}$ is Irrational when $p$ is Prime

I have found various proofs that $\sqrt{p}$ is irrational on this site, but I didn't find one similar to the one that I am about to post, so I am wondering if it is free of logical problems. Here is ...
0
votes
0answers
26 views

How to calculate the length of this plane curve (loxodrome/rhumb line)?

I am trying to calculate the length of a (what I believe is) a loxodrome, using differential geometry. I am given a curve $\gamma(t)=\big(\theta(t),\varphi(t)\big)\subset \mathbb S^2$ that ...
3
votes
0answers
27 views

Properties of the Discrete Logarithm Problem

I am self-studying Hoffstein's An Introduction to Mathematical Cryptography, and this is problem 2.3 (p. 107-08). Let $p$ be a prime and $g$ an element in $\mathbb{F}_p^*$ with order $r$. ...
0
votes
1answer
18 views

Prove regular expression with induction

I need help proving the following regular expression via induction. I have the base case (easy of course) but I'm having a difficult time determining the inductive case. A regular expression over ...
3
votes
3answers
84 views

Solve the equation $7t+[2t] =52 $ ,where $[x]$ denotes the floor function for $x$.

Solve the equation $7t+\left\lfloor 2t\right\rfloor =52 $. My effort Using the fact that for any number $x$ we have that $x=\left\lfloor x\right\rfloor+\{x\}$ (where $\{x\}$ is the fractional ...
0
votes
1answer
29 views

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups. Suppose $x=ab,a\in H\times 1,b\in 1\times K$ Then $x=(h,1)(1,k)$ where $h\in H,k\in K$ Hence $x=(h,k)\in H\times K$ Let ...
0
votes
2answers
52 views

Proof Verification: If $x$ is a nonnegative real number, then $\big[\sqrt{[x]}\big] = \big[\sqrt{x}\big]$

Let $x$ be a nonnegative real number and denote $[x]$ as the greatest integer less than or equal to $x$. We will attempt to prove that $\big[\sqrt{x}\big] = \big[\sqrt{[x]}\big]$. First suppose that ...
0
votes
0answers
51 views

Hypothetical proof of Goldbach's conjecture? [on hold]

Goldbach's conjecture: Every even number greater than 4 is the sum of two prime numbers. An equivalent statement is; For all $n\geq 2$ there exists a number $e(n)$ such that $n-e(n)$ and ...
3
votes
1answer
36 views

Proof Verification: Show that $\Big[\frac{x+n}{m}\Big] = \bigg[\frac{[x]+n}{m}\bigg]$

Let $m,n \in \mathbb{Z}$ and let $x \in \mathbb{R}$. Let $[x]$ denote the floor function. We will attempt to prove $$\Big[\frac{x+n}{m}\Big] = \bigg[\frac{[x]+n}{m}\bigg]$$ Suppose without loss of ...
3
votes
2answers
85 views

Prove that $\lim_{x\rightarrow \infty} \frac{x^2 - 1}{x^2 + 1} = 1$ using definition of limit.

Ok, so if I have to use definition, then I should prove something like this: $(\forall \epsilon >0)(\exists k>0)(\forall x \in X)$ then if $ x>k$ then $|f(x) - L| <\epsilon$ $L$ is ...
1
vote
2answers
48 views

If $\frac{ay-bx}{p}=\frac{cx-az}{q}=\frac{bz-cy}{r} $

If $$\frac{ay-bx}{p}=\frac{cx-az}{q}=\frac{bz-cy}{r}$$, prove that : $$\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$$ My solution, $$\frac{c(ay-bx)}{cp}=\frac{b(cx-az)}{bq}=\frac{a(bz-cy)}{ar}$$. Now, I could ...
1
vote
2answers
26 views

Product of terms involving complex exponents [on hold]

I have worked out the $\prod_{k = 1}^{50}i^k$. I get answer is $-i$. Is it correct?
0
votes
2answers
53 views

Prove or disprove: For non-negative integers $m$ and $n$, $m!n! = (mn)!$

I have rewritten the question as "If $m$ and $n$ are non-negative integers, then $m!n!$ = $(mn)!$" Here is my current attempt. I am not sure if I am on the right path. Proof. Let $m$ and $n$ be ...
-4
votes
4answers
74 views

Find $A$ and $B$ such that $A⊈B$ and $B⊈A$? [on hold]

I need to prove that the subset relation “$⊆$” on all subsets of $\mathbb Z$ is not a total order and I'm going to do this by finding $A$ and $B$ such that $A⊈B$ and $B⊈A$? Is there a simple solution ...
0
votes
1answer
71 views

Does this prove the sequence $5+(-1)^n$ does not have a limit?

The question is "Consider the sequence $s_n=5+(-1)^n$. Prove that this sequence does not have a limit". My professor in class proved this by choosing $n_1$ to be even, $n_2$ to be odd, and ...
21
votes
7answers
3k views

What is the flaw of this proof (largest integer)?

Let $n$ be the largest positive integer. Since $n ≥ 1$, multiplying both sides by $n$ implies that $n^2 ≥ n$. But since $n$ is the biggest positive integer, it is also true that $n^2 ≤ n$. It follows ...
3
votes
3answers
47 views

Find the area of the region described by $|5x|+|6y| \le 30 $

Find the area of the region described by $|5x|+|6y| \le 30 $ (where $|z|$ denotes the absolute value of $z$). My effort Imagining a number line and interpreting the problem as the request to ...
0
votes
1answer
44 views

Using pumping lemma

I'm trying to prove that the language $\mathcal L = \{w \in \{0,1\}^* ∣ w \leq w′ \text{ where }w′ \text{ is any rotation of }w\}$ is not a regular language. Note: The inequality is with respect to ...
2
votes
1answer
69 views

Would you confirm this as a proof to the Pythagorean theorem?

I'm new in mathematics, and trying to build my way up starting by doing simple tasks. My current one is proving the Pythagorean theorem without looking it up. This photo contains my current "proof" ...
1
vote
1answer
20 views

Image of Upper Half Disc under $w = 1/z$

I need to find the image of the upper half disc $|z|<1$, $Im\, z >0$ under the inverse transformation $w = 1/z$. Now, since $|z|<1$, $|z|^{2}<1$. Rewriting this as $z\overline{z}<1$, ...
0
votes
1answer
20 views

$A_n$ is generated by 3-cycles given $n\geq 3$. Is this proof correct?

The elements of $A_n$ is either of the form $(a,b,c,...)...$ or of the form $(a,b)(c,d)...$ In both cases, the element is a product of an even number of transpositions, not pairwise disjoint in the ...
0
votes
1answer
36 views

Proving $\mathbb{R}/\sim$ is homeomorphic to unit circle

Let $S$ be the unit circle in $\mathbb{C}$, standard topology. Define the equiv. rel. $\sim$ on $\mathbb{R}$ as $x\sim y\iff x - y\in\mathbb{Z}$. I would like to prove that $\mathbb{R}/\sim$ is ...
0
votes
0answers
30 views

Quick proof check [on hold]

Intro to analysis by gaughan 1.4 #45. Show that if x is any real number, there is a sequence of rational numbers converging to x.
1
vote
2answers
16 views

Prove that the space $\Bbb R_K$ is not regular.

Prove that the space $\Bbb R_K$ is not regular. where the basic open sets on $\Bbb R_K$ is given by $\{(a,b):a,b\in \Bbb R\}\cup \{(a,b)-K\}$ where $K=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$. ...
0
votes
0answers
19 views

How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...
2
votes
1answer
55 views

If $x$ is an isolated point of $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$. [duplicate]

Is the following proof valid? (Note: I know there is a post discussing this problem, but I am curious to see if my argument works). This problem is different from another post that is similar with ...
1
vote
1answer
21 views

Transformations and Dependence

Hi, for these problems I generally get the gist of it. If you have some linearly dependent vectors $v_1, \ldots, v_m$ in $\mathbb{R}^n$ then when you transform those vectors $T(v_1), \dots, T(v_m)$ ...
-1
votes
0answers
18 views

renewal process and probability(exercice) [on hold]

Let $N_t$ a renewal process. Let $A_t=t-S_{N_t-1}$, $S_{N_t}=X_1+...+X_{N_t}$ with $X_i$ the jumps moments. Let $Z_A(t)=P(A_t \leq u)$ 1) How to show $P(A_t \leq u |X_1=x)=P(A_t \leq u |X_1 \geq t)$ ...
5
votes
0answers
35 views

Find the inverse of the following piecewise defined function

Find the inverse of $f$ if $f(x)=$ $$ \begin{cases} \sqrt{2-x}, &\text{for $x<0$}\\ 1-x^2, &\text{for $x \ge 0$} \\ \end{cases} $$ My effort For $y=\sqrt{2-x}$ ,we find ...
1
vote
0answers
19 views

A Possible Logical Problem with Showing that $x$ is a Boundary Point Whenever It is an Isolated Point

Prove: If $x$ is an isolated point of a set $S$, then $x \in \mathrm{bd} \, S$. I have two ways to solving this problem, but I believe the first one has a logical issue which I will explain below. ...