For questions about the formulation of a proof, not about the mathematics behind it.

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0
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5answers
44 views

Proof by Mathematical Induction for all natural numbers n.

$1^3 + 2^3 + \cdot \cdot \cdot+ n^3 = $ $[ \frac{n(n+1)}{2}]^{2} $ $\text{My question for this problem is that I got stuck at a certain point}$ $\text{and I do not know where to go. This is what I ...
1
vote
2answers
46 views

Help determining if a finite subset of $\mathbb R$ is closed and bounded.

If $\{A_n \; : \; n \in \mathbb N\}$ is any collection of subsets of $\mathbb R$, with each set $A_n$ containing finitely many numbers, then the union $\bigcup_{n=1}^{\infty}A_n$ is closed and bounded....
2
votes
3answers
67 views

The limit of $f(x,y)= \dfrac {x^2 y}{x^2 + y^2}$ as $ (x,y) \to (0,0)$

In order to prove that the limit as $(x,y)$ approaches to $(0,0)$ of $f(x,y)= \dfrac {x^2 y}{x^2 + y^2}$ is equal to $0$ is wanted to proof: for ever $\beta\gt0$ exists some $\delta\gt0 $such that ...
0
votes
0answers
19 views

About continuity of scalar fields.

Using the usual definition of limits, with "epsilon and deltas", how can I show that if $x=(x_1,\dots,x_n)$ is a vector in $R^n$, and $f\colon J\to R$,where $R$ is the set of real numbers and $J$ is a ...
-1
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0answers
28 views

Determine if the following sentence is a proposition

$2^{101}-1 $ is a prime. Besides being a prime and not being a prime, is there any other case the answer could be? If there isnt a third case, Then is a proposition correct? but if there is then is ...
0
votes
1answer
35 views

Prove that $x=5\cos(6x)$ for some $x$ in the interval $\left(\frac{\pi}{6},\frac{\pi}{3}\right)$

Prove that $x=5\cos(6x)$ for some $x$ in the interval $\left(\frac{\pi}{6},\frac{\pi}{3}\right)$ using the the IVT I'm not entirely sure how to prove this but I set it up this way: $f(x)=x-5\cos(...
0
votes
1answer
17 views

Sandwiched sequence converge to same limit in $\omega_1$

I am stuck on a question that might need a trick to crack, any help is appreciated Problem statement Let $(a_n), (b_n)$ be sequences on $\omega_1$ as a topological space, such that $a_n \leq ...
1
vote
1answer
42 views

Prove using the epsilon definition

I'm trying to prove the below using the $\epsilon$ definition: $\epsilon$-definition: $\;\left|s_n-s\right| \lt \epsilon$ $\lim \limits_{n \to \infty}\frac{(-1)^n\cos \sqrt{n}}{\sqrt[3]{n}}=...
2
votes
3answers
45 views

How to show that the set of all $(x,y)$ such that $3x^2 + 2y^2<6$ is an open set.

How can it be proved that the set of all $(x,y)$ such that $3x^2 + 2y^2<6$ is an open set? I tried to prove directly the aforementioned statement. Without success I tried to prove that the image ...
0
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0answers
20 views

After the midnight when all trhee clock-hands are in same direction (superposed) again?

in order to solve this question: "After the midnight when all trhee clock-hands are in same direction (superposed) again?" that appears to be simple, probably is, but i could not give a answer ...
1
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3answers
41 views

Prove that there exists n consecutive composite numbers

I'm asked to prove that there exists n consecutive composite numbers. This is what I've come up with. $$n! + 1 = (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot \dotsc \cdot n) + 1 $$ is a prime number ...
3
votes
2answers
84 views

Prove that $\lim \limits_{x \to 5}\left(4x^2-7\right)=93$

So I first need to determine the limit and then prove it: $\lim \limits_{x \to 5}\left(4x^2-7\right)$ So $L=93$ And thus $\left|f(x)-L\right|=\epsilon$ and $\left|x-c\right|=\delta$ Plugging ...
0
votes
1answer
22 views

How to prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference?

how can i prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference? i mean supose that $A=(a_1,...,a_m)$ and $B=(b_1,...,b_m)$ both ...
3
votes
1answer
28 views

Prove that if $f(x,t)$ is continuous in $D=\{(x,t):x\in[a,b]\land t\in[c,d]\}$ then $F(x)=\int_c^d f(x,t)\mathrm dt$ is continuous in $[a,b]$

Prove that if $f(x,t)$ is continuous in $D=\{(x,t):x\in[a,b]\land t\in[c,d]\}$ then $F(x)=\int_c^d f(x,t)\mathrm dt$ is continuous in $[a,b]$ This is about Riemann integration. I dont know how to ...
2
votes
1answer
40 views

Proof verification about inverses of linear mappings.

In order to prove the following statement: "Let $F: U\to V$ be a linear map, and assume that this map has an inverse mapping $G\colon V \to U$. Then $G$ is a linear map." In Serge Lang book he ...
0
votes
1answer
35 views

Proving that the following is a convex subset

It is given that $I(S)=\{z|\exists x,y \in \mathbb{R}$ such that $(x,y,z)^t \in S$}$\subseteq \mathbb{R}$. How do I show that I(S) is a convex subset? We know that $S \subseteq \mathbb{R}^3$ is ...
0
votes
1answer
38 views

Proof of non-constant analytic functions

Use the following theorem: "A function that is analytic in a domain $D$ is uniquely determined over $D$ by its values in a domain, or along a line segment, contained in D" to show that if $f(z)$ ...
0
votes
1answer
35 views

How can i proof that every high order derivative of $\frac{1}{1+x}$ is equal to$ (-1)^kk!$ at point $0$.

In order to calculate the Taylor-Maclaurin polynomial $\frac{1}{1+x}$ of order $n$ at point $0$, i used the identity: $$\sum_{i=0}^n x^i + \frac{x^{n+1}}{1+x} = \frac{1}{1+x}\qquad (I)$$ and i ...
1
vote
0answers
33 views

Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
0
votes
1answer
24 views

Prove if $x_1,…,x_n$ are natural numbers with $n\geq2$ then $x_1x_2…x_n$ is odd iff $x_i$ is odd for all $i$, $1\leq i\leq n$

I am not sure if Im on the right track here but if any one could help out I would greatly appreciate it. Prove if $x_1,...,x_n$ are natural numbers with $n\geq2$ then $x_1x_2...x_n$ is odd iff $...
0
votes
0answers
28 views

How to do proofs involving sets

I have just recently started preparing for a course I will be taking next year, but I have very limited knowledge as it relates to proofs. It seems as though the only proofs I am slightly familiar ...
1
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1answer
32 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
1
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2answers
51 views

Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
0
votes
0answers
35 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
1
vote
1answer
76 views

Formulating a problem in terms of set theory

Here is one problem I was trying to solve just by trial-and-error method. However, I was thinking about how to write the clear solution using set theory. Problem: A notebook contains exactly $100$...
0
votes
2answers
41 views

binomial inequality with sums

Assume I have a series of numbers $a_1 \dots a_n$ where $0 \leq a_i \leq n-1$ and a positive integer $r$. how to show that the sum of number of ways to choose $r$ from $a_i$ is at least as $n$ times ...
1
vote
1answer
35 views

Is it sufficient to prove that a function is an open map by looking at the basis element?

I am trying to prove that the projection map $\pi_X:(X, T)\times (Y,J) \to X$ is an open map But I don't know if I can use the basis element directly, so my proof is quite round about and lengthy ...
1
vote
1answer
16 views

Show that there exists at most one extension of $f$ whose co-domain is a Hausdorff space [duplicate]

I want to show the following Suppose $A \subset X, f: A \to Y$ is continuous, $Y$ is Hausdorff. Show that there is at most one continuous extension $g: \overline A \to Y$ I feel like I am ...
1
vote
2answers
46 views

How to see that $f(t) = (t, 2t, 3t, \ldots)$ continuous in the product topology

I am trying to check whether $f: \mathbb{R} \to \mathbb{R}^\omega$ $f(t) = (t, 2t, 3t, \ldots)$ is continuous or not in the product and box topology. But I have a feeling I don't have the ...
3
votes
3answers
69 views

Topology: is it ever good to write $x \in U \in \mathfrak{T}$

Sometimes I come across a sentence in my topology book that says, let $U$ be an open set that contains $x$ I can't help but write it down as: Let $$x \in U \in \mathfrak{T}$$ Is it good ...
1
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1answer
39 views

Fisher information for exponential family: Regularity conditions

for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus....
3
votes
4answers
93 views

$a^n$ even implies $a$ even

I've tried to prove that $(\forall a,n>0 \in \mathbb{N}),(a^n \text{ even} \implies a \text{ even})$, can someone tell me whether my proof is sound? Lemma 1: $a \text{ even} \implies a^2 \text{ ...
3
votes
1answer
65 views

proof of Triangle Removal Lemma

Where can I find a proof of the following version of Triangle Removal Lemma (or any version equal to it)? Let $G(V,E)$ be a graph on $n$ vertices such that it contains $\varepsilon n^3$ triangles, ...
0
votes
1answer
14 views

If unions of two families sets are disjoint then families of sets are disjoint too.

I have read that theorem "Suppose $\mathcal{F}$ and $\mathcal{G}$ are families of sets. If $\cup\mathcal{F}$ and $\cup\mathcal{G}$ are disjoint, the so are $\mathcal{F}$ and $\mathcal{G}$" is ...
2
votes
2answers
43 views

Proof verificication and question of rigour: $A$, $B$, connected implies union is connected

Don't mark this as duplicate. The other question does not help me figure out how rigorous MY proof is. Problem: Let $A$ and $B$ be connected subsets of a metric space and let $A\cap\overline{B}\neq\...
0
votes
1answer
44 views

Nice way to prove a limit.

I know how to prove the following limit $$\lim _{\epsilon \rightarrow 0} \frac{a^{\epsilon}-1}{\epsilon}=\ln(a)$$ But I am looking for a nice way to do it, a little elegant. Would you have one?
1
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2answers
57 views

Prove that the directrix-focus and focus-focus definitions are equivalent

(NOTE: This is my attempt at answering this question and this question, but I rewrote it in order to make it easier for me to solve. Also, I've made a YouTube video explaining this whole proof. Note ...
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3answers
63 views

Prove $\forall n \in \mathbb{N}$ where $ n \neq 1 , n + \frac{1}{n} > 2$ using completing the square.

I have got this far; I am only unable to understand how to finish the proof. $n>0 \implies n + 1/n > 0 \implies n + 1/n + 2 - 2 > 0 \implies {\big(\sqrt{n}+\frac{1}{\sqrt{n}}\big)}^2 - 2 >...
-4
votes
3answers
74 views

prove that the $5$th root of $r$ is irrational if $r$ is irrational [closed]

I am trying to learn mathematics for computer science in own efforts. I got this task to prove that $\sqrt[5]{r}$ is irrational, given that $r$ is irrational. Normally if I had to prove that $\sqrt{2}...
3
votes
1answer
124 views

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable?

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable? To clarify: the problem stated that the composition is well ...
1
vote
2answers
35 views

Show that a set is open if and only if each point in S is an interior point.

I am in a complex analysis class and have been asked to prove this. I know I have to prove both ways so. If a set is open then each point in $S$ is an interior point. Proof: Let $S$ be an open set,...
2
votes
1answer
122 views

How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? [duplicate]

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function... I tried my hand at the ...
4
votes
2answers
88 views

problem proving: $(1+q)(1+q^2)(1+q^4)…(1+q^{{2}^{n}}) = \frac{1-q^{{2}^{n+1}}}{1-q}$

I'm trying to prove this, and it is really frustrating, because it seems a really easy problem to prove, however, I'm having a little problem with exponents: $$(1+q)(1+q^2)(1+q^4)...(1+q^{{2}^{n}}) = ...
0
votes
1answer
59 views

Proof that $|S| \leq |T|$ if $S \subseteq T$.

Let $S$ and $T$ be sets. I am having trouble proving that if $S \subseteq T$, then $|S| \leq |T|$, where $|S|$ is the cardinality of $S$.
2
votes
6answers
101 views

Three positive numbers a, b, c satisfy $a^2 + b^2 = c^2$; is it necessarily true that there exists a right triangle with side lengths a,b and c?

If so, how could you go about constructing it? If not, why not? I am new to proofs and I was reading a book where they posed this question. I understand that if we are given any right triangle, the ...
0
votes
2answers
47 views

(Real Analysis) Topology: Prove $f(cl S)\subseteq clf(S)$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Show: $f(\overline{S})\subseteq \overline{f(S)}$ for $S\subseteq \mathbb{R}$ (Note: $\overline{S}$ denotes the closure of S; $\partial S$ ...
0
votes
2answers
54 views

(Rigor/Validity of Proof) Every sequence of reals in a compact set has a convergent subsequence

[ADDED/MODIFIED]: I began my proof with a compact set, but this was a wrong start. Although the comments are valid, I should've started with a bounded set. Because what I want to establish first is ...
0
votes
2answers
20 views

Strong Induction issue

I am trying to prove a statement using strong induction but I seem to be getting stuck. I don't know if did something wrong or I am just not recognizing an opportunity for factoring/how to factor ...
1
vote
2answers
65 views

Proof that $\overline{P(z)} = P(\overline{z})$ for polynomial $P$ with real coefficients

Let $$ a_0, a_1, a_2, a_3, \ldots , a_n \quad (n \ge 1)$$ denote real numbers, and let $z$ be any complex number. With the aid of $$ \overline {z_1 +z_2+ \ldots +z_n} = \overline z_1 +\overline z_2+ \...
0
votes
2answers
46 views

Easy proof the set of finite Set in countable is countable [duplicate]

Suppose I know a result that the set of finite sets in $\mathbb{N}$ is countable. Is there a very quick way to show that the set of finite sets in any $X$ countable is countable? Idea...two sets ...