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Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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1answer
17 views

Prove $\lim\limits_{x\to\frac52}\frac1{4x-8}=0.5$ via delta epsilon

prove $\lim\limits_{x\to\frac{5}{2}}\frac{1}{4x-8}=0.5$ via delta epsilon I have the following $|\frac{1}{4x-8}-\frac{1}{2}|<\epsilon$ $|\frac{5-2x}{4x-8}|<\epsilon$ which I got after ...
0
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0answers
14 views

How to prove $B = C$ If $A \cap B = A \cap C$ and If $A \cup B = A \cup C$ [duplicate]

I know that I need to prove $B \subseteq C$ and $C \subseteq B$ my try(of proving $B \subseteq C$): Suppose $A \cap B = A \cap C$. Futher suppose that $A \cup B = A \cup C$. Futher suppose $x \in B$ ...
3
votes
3answers
38 views

Proving $x^2 - 4y^2 = 7$ has no natural numbers

Ok so I needed to prove this by contradiction. Let $P:~x^2 - 4y^2 = 7$ and $Q:~x,y$ are not natural numbers Note that $N$ does not include $0$ OK to begin to prove by contradiction we are given $P$...
1
vote
1answer
14 views

Proof that solutions to Cauchy F.E. over $\mathbb{Q}$ are linear

I would like to prove that the solutions to Cauchy's functional equation over $\mathbb{Q}$ are linear, that is, all solutions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ have the property $f(x)=cx$ for some $...
1
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0answers
8 views

Show that a function cannot be expressed as a strictly increasing transformation of another function

Consider a function $\Phi: \mathcal{I}\subseteq \mathbb{R}\rightarrow \mathbb{R}$. Suppose (a) $\Phi(0)=0$ (b) $\Phi(1)=1$ with $0\in \mathcal{I}$ and $1\in \mathcal{I}$. Consider a function $\...
1
vote
0answers
15 views

Deriving an inequality related to an induced matrix norm

Suppose we have the induced matrix norm for an arbitrary $n\times n$ matrix $A$ given by, \begin{equation*} |||A|||_{\infty,h} = \max_{1\leq i \leq N}\left(\sum_{j=1}^N |A_{ij}|\right). \end{equation*...
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0answers
27 views

Showing a Set is a Partition of Another

Here's a homework question I'm probably over thinking. Prove that if $S$ is a set and $A$ is a nonempty proper subset of $S$, then $\Pi = \{A,S\setminus A\} $ is a partition of $S$. The claim is ...
1
vote
2answers
17 views

Is the sequence of real number that are $0$ from some point Sense in the box or product topologies

Let $A=\{(x_n)_{n\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}|\exists M\in\mathbb{N} ,\forall n>M, x_n=0 \}\subset\mathbb{R}^\mathbb{N}$, series of real numbers that are zero from some point forward. ...
1
vote
2answers
45 views

Proving $A\subset B\implies |A|\leq |B|$

I would like to prove the following: Let $A,B\subset\mathbb{R}$ be non-empty finite sets. Prove that if $A\subset B$, then $|A|\leq |B|$. We are also given the following theorem: Theorem $1....
0
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1answer
30 views

Show that $\|b \|^2\ (\sin\theta)^2 = \min_x\ \|ax-b\|^2$ where $\theta$ is the angle between $a,\ b$.

Show that $\|b \|^2\ (\sin\theta)^2 = \min_x\ \|ax-b\|^2$ where $\theta$ is the angle between $a,\ b$. This is for vectors $a, b \in \mathbb{R}^m$ \ {$0$}. The second part of the question says: ...
0
votes
0answers
14 views

Internal stability of a discrete-time system

These are two parts of a much larger proof I'm working on, can't figure how ii implies iii though. $x(k+1)=Ax_k,x(0)=x_{0}$ Where $A∈\mathbb{R}^{n×n}$ is a real constant matrix. i) All the ...
0
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0answers
19 views

How do you observe continuity in the following piecewise function?

Consider the piecewise function: $$f(x) = \begin{cases} x, & \text{if x is} \in\mathbb{Q} \\ 0, & \text{if x is} \notin\mathbb{Q} \end{cases}$$ I am asked to define f(0) such that f(x) is ...
0
votes
1answer
81 views

Proof of this integral identity $\int_0^a{\frac{\tan^{-1}{\left(\sqrt{\frac{2a^2-x^2}{4a^2-x^2}}\right)}}{\sqrt{4a^2-x^2}}dx} = \frac{\pi^2}{32}$

I would like to proof this identity I conjectured $\forall a\in \mathbb{R}^+$ $$\int_0^a{\frac{\tan^{-1}{\left(\sqrt{\frac{2a^2-x^2}{4a^2-x^2}}\right)}}{\sqrt{4a^2-x^2}}dx} = \frac{\pi^2}{32}$$ My ...
0
votes
2answers
27 views

In a simple example, why the set $S$ is unbounded?

There is a real number $y$ such that $y>\dfrac {1}{1+x^2}$ for any real number $x$. Prove or disprove this question. Answer. Let $S=\left\{ \dfrac {1}{1+x^2}:x\in\mathbb{R}\right\}$. So since $S=(...
0
votes
1answer
23 views

Prove that if $a_n$ tends to $\infty$ then $\frac{1}{a_n}$ tends to 0

I have the definition that a sequence to tends to $\infty$ if, for every $C>0$, there exists a natural number $N$ such that $a_n > C$ for all $n>N$ And I have the definition that a sequence ...
0
votes
1answer
24 views

What would a rigorous proof that the vector space of functions from $\mathbb{Z}$ to $\mathbb{R}$ is not finite-dimensional look like?

So consider the vector space of functions from $\mathbb{Z}$ to $\mathbb{R}$. Intuitively, this vector space is not finite-dimensional. Indeed, $\mathbb{Z}$ and $\mathbb{N}$ are the same by "zig-...
0
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2answers
16 views

How to prove $(A \cup (B - C)) \cap B^c \subseteq A - B$

My try: Suppose $(A \cup (B - C)) \cap B ^c$. We know $x \in A \cup (B - C)$ and $x \in B^c$. So, $x \in A$ or $x \in B - C$ and $x \in B^c$. From there, I see two possible cases: 1 - $x \in A$ and $...
3
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0answers
31 views

I am writing a formal proof for the first time

I have written a formal proof for the first time and am looking for input on any errors I may have made or general tips thanks. Theorem: For all real numbers x and y, $$x^2 + y^2 + 1 \...
2
votes
1answer
18 views

Is this proof of $A \cup (B - (A \cap B)) \subseteq A \cup B$

My try: Suppose $x \in A \cup (B - (A \cap B))$. If $x \in A$, then $x \in A \cup B$. If $x \in B - (A \cap B)$, then $x \in B$ and $x \notin A \cap B$. Specificaly, $x \in B$. Hence $x \in A \...
1
vote
0answers
10 views

Let $t_n$ be a convergent sequence and suppose that $\lim(t_n)>a$. Prove there exists a number $N$ such that $n>N$ implies $t_n>a$.

My start: Say $\lim(t_n)=b$ and let $\epsilon>0$. Then there exists $N$ such that $n>N$ implies $|t_n-b|<\epsilon$. or should I say: Let $a>0$. Then there exists $N$ such that $n>N$ ...
0
votes
0answers
21 views

Help with another measure theoretic exercise [duplicate]

I really need friends interested in Analysis with whom I could talk about problems lol So, the setting is the following. First of all, by convention, $0 \cdot \infty = 0$. Let $m$ be the Lebesgue ...
0
votes
1answer
23 views

proof on equality of sets

If i want to Prove $A^c \cup B^c$ = $(A \cap B)^c$ by a string of equalities =$\{x|x\in A^c \cup B^c\}$ =$\{x|x\in A^c orx\in B^c\}$ =$\{x|x\notin A orx\notin B\}$ =$\{x|x\notin (A \cap B)\}$ =$\{...
-1
votes
0answers
24 views

Proof for the transitive property of inequality [on hold]

How to prove that $$ \forall x, y \in \mathbb{R}. x > y \Leftrightarrow (\forall z \in \mathbb{R}. z \ge x \Rightarrow z > y) $$ is true using quantified statements?
2
votes
1answer
34 views

Does this follow from congruence?

Let a and b be two distinct prime numbers and x and y are integers. Is the following true? ($x \equiv y \mod a$) and ($x \equiv y \mod b$). So, $a|(x-y)$ and $b|(x-y)$. This means $x-y=ab\phi$ with $\...
1
vote
1answer
29 views

General term for the sum $\sum \sin(k)$ [duplicate]

How do I prove that: $$\sin(1)+\sin(2)+\cdots+\sin(n)=\frac{\sin\left(\frac{n+1}2\right)\sin\left(\frac n2\right)}{\sin\left(\frac12\right)}?$$ I have tried to to use the formula $\sin(2x)=2\sin(x)\...
-1
votes
1answer
25 views

Parameters A and B must be determined such that for each x∈ℝ \ {1, -1, -2} the following equation is satisfied: [on hold]

$$\frac{(7x+2)(x+1)}{(x^2-1)(x+2)} = \frac A{x−1}+\frac B{x+2}$$ I never did such a task before. My first idea would be to shorten the left side by (x+1), which would result in. $$\frac{7x+2}{(x^2-...
4
votes
1answer
55 views

Prove $\lim_{x \rightarrow 3}x^2=9$ via delta epsilon

Prove $$\lim_{x \rightarrow 3}x^2=9$$ via delta epsilon This is what I have so far $|x^2-9|<\epsilon$ and $0<|x-3|<\delta$ Let $z=x-3$. $|(z+3)^2-9|<\epsilon$ $\delta(\delta+6)<\...
1
vote
1answer
23 views

Stuck on a term in $Var[\hat{\beta_o}]$ Proof

So I was trying to prove that $Var[\hat{\beta_o}]=\dfrac{\sigma^2n^{-1}\sum{(x_i)^2}}{\sum{(x_i-\bar{x}})^2}$ And I got stuck with the part $\dfrac{ -2\bar{x}}{\sum{(x_i-\bar{x})^2}}\sum[{(x_i - ...
0
votes
2answers
28 views

Differences of between ''either $f(x)=0$ for all $x$ or $g(x)=0$ for all $x$'' AND ''for all $x$ either $f(x)=0$ or $g(x)=0$''

Let $f,g$ be two functions from $\mathbb{R}$ to $\mathbb{R}$. Assume for all $x$ $f(x)g(x)=0$, then i) then either $f(x)=0$ for all $x$ or $g(x)=0$ for all $x$. ii) then for all $x$ either $f(x)=0$ ...
12
votes
1answer
264 views

İMO 2011: Prove that, for all integers $m$ and $n$ with $f(m)<f(n)$, the number $f(n)$ is divisible by $f(m)$

Problem: Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m-n)$. Prove ...
-2
votes
1answer
37 views

How can I prove that following Statements are true or false?

I have to decide whether the following statements are true by using a proof or a counter-example. $$\begin{align} &(\text{i})~~~ a < b \text{ and } c < d \Rightarrow ~a - c < b - d\\ &...
0
votes
3answers
37 views

If natural number $n$ doesn't has the form of $k^{m}$, for natural $k$ and $m$, then $x^{m}=n$ has no rational root.

Prove: If a natural number $n$ doesn't has the form of $k^{m}$, where $k$ and $m$ are natural numbers, then the equation $x^{m}=n$ has no rational root. How do I start to prove with contradiction (or ...
1
vote
1answer
31 views

Proving that $(a,b)$ is $F_{\sigma},\forall a,b\in\mathbf{R}$

I am required to prove that the interval $(a,b)$ is a $F_{\sigma}$-set i.e. it can be written as a union of countably many closed sets in $\mathbf{R}$. The following is my attempt so far. I did ...
4
votes
1answer
49 views

Not sure how to solve $\lim_\limits{x\to0}{{\sqrt{x^2+1}-1}\over\sqrt{x^2+16}-4}$

So I got this problem: Determine the following limit value: $$\lim_\limits{x\to0}{{\sqrt{x^2+1}-1}\over\sqrt{x^2+16}-4}$$ What I tried is: $\large{\lim_\limits{x\to0}{{\sqrt{x^2+1}-1}\over\...
-1
votes
1answer
18 views

justification of simple idea involving union

How would you justify going from $x\notin$ $A\cup B\ $ $x\notin A$ and $x\notin B$ in a proof, where the same values of x satisfy both statements. would by definition of U be sufficient or would ...
0
votes
5answers
41 views

differences of ''and'', ''or'' in the questions

Prove or disprove in the below questions: $1.$ Let $a$ and $b$ be real numbers. Then $(a+b)^3=a^3+b^3$ implies $a=0$ OR $b=0$. Disproof. Let $a=-1$ and $b=1$. Then, we also get $(a+b)^3=a^3+b^3=0$....
1
vote
1answer
76 views

Finding range of expression $f(x,y)=x^2+y^2$

Finding range of $f(x,y)=x^2+y^2$ subjected to the condition $2x^2+6xy+5y^2=1$ without Calculus Try: Let $k=x^2+y^2,$ Then $\displaystyle k=\frac{x^2+y^2}{2x^2+6xy+5y^2}$ Now put $\...
4
votes
2answers
99 views

Commutative Semigroup

Let $S$ be a Semigroup with the two following properties, $(1):$ for all $x$ in $S$ we have $x^3=x$ $(2):$ for any $x,y$ in $S$ we have $xy^2x=yx^2y$. Then prove that this Semigroup $S$ is ...
1
vote
1answer
39 views

Prove by induction that $W_n = F_{2n+2}$

My problem relies on an earlier recursive definition that we solved in class: $W_n = 3W_{n-1}- W_{n-2}$ if $n \ge 2, W_0=1$, and $W_1=3.$ It also recalls the Fibonacci recursive definition of $F_n = ...
5
votes
2answers
282 views

Find range of exponent

I have the following function I need to find the range for and I'm not sure if I'm on the right direction. $f(x,y) = e^{-x^2-(y-1)^2}$ $x$ & $y$ are real-numbers. I'm thinking that the range is ...
3
votes
1answer
29 views

Differential Equations proof for Prove that $\dim(\ker(TU)) ≤ (\dim\ker(T)) + \dim(\ker(U))$.

Let $T$ and $U$ be linear transformations $V → V$ with finite-dimensional kernels. Prove that $\dim(\ker(TU)) ≤ (\dim\ker(T)) + \dim(\ker(U))$ My tutor suggested that I create two new ...
0
votes
2answers
42 views

Show that $Ax^2+Bx+C>0 $ for all real $x$ if and only if $A>0$ and $B^2-4AC<0$.

Show that For all real $x$ , $Ax^2+Bx+C>0 $ if and only if $A>0$ and $B^2-4AC<0$. Case 1 : Suppose that $A>0$ and $B^2-4AC<0$. Let $y=Ax^2+Bx+C$. Then $Ax^2+Bx+(C-y)=0$ ...
0
votes
1answer
47 views

show that $\operatorname{int}(A)$ $\subset $ $A$ $\subset$ $\overline{A} $ [closed]

$$\operatorname{int}(A) \subseteq A \subseteq \overline{A}$$ need help to prove this statement. Suppose $\operatorname{int}(A)$ is open show that $\operatorname{int}(A)$ is a subset of A is a subset ...
1
vote
2answers
69 views

$a + b = 2$ implies $a^c + b^c \ge 2$ for any real $c \ge 1$

If $a, b, c$ are positive reals such that $c \ge 1$ and $a + b \ge 2$ then $a^c + b^c \ge 2$. Is there any elementary way to prove it without using calculus and advanced inequalities like Jensen's? ...
-1
votes
1answer
56 views

How to prove that if $\lim(a_n)=a$ then $\lim(\frac{1}{a_n})=\frac{1}{a}$?

How to prove that if $\lim(a_n)=a$ then $\lim(\frac{1}{a_n})=\frac{1}{a}$ ? So far I have the following written down $|a_n-a|<\epsilon$ $|\frac{a}{a_n}-\frac{1}{a}|=\frac{|a_n-a|}{a_na}$ and ...
0
votes
0answers
19 views

proof involving double complement [duplicate]

Prove $(A^c)^c$ = $A$ $(A^c)^c = \{x|x\in (A^c)^c\}$ $(A^c)^c = \{x|x\notin A^c\}$ by definition of complement $(A^c)^c = \{x|\ (x \in A)\}$ by definition of complement Therefore $(A^c)^c=A$ ...
0
votes
3answers
46 views

How do I prove directly from the $\delta$-$\epsilon$ definitions that if $\lim_{x\to 0} f(x) = \infty$, then $\lim_{x\to 0} f(x)$ does not exist?

The definition of $\lim_{x\to 0} f(x) = \infty$ is: $\forall N\in\mathbb{R}\;\exists \delta\gt 0\; (0<\lvert x\rvert<\delta\implies f(x)>N)$ The definition of "$\lim_{x\to 0} f(x)$ does not ...
1
vote
0answers
18 views

Proving the existence of real solutions to a system of multivariate polynomials for a range of parameters

I need to prove the existence of real solution(s) to a system of 7 polynomial equalities in 14 variables up to degree 6 for a range of values of 5 real parameters $t_1,t_2,t_3,x_0,x_3$, with $t_1>0,...
0
votes
1answer
16 views

Proving unit relation features [closed]

I am trying to prove the following unit relation features: $R ⊆ X × X$, $IdX ◦ R = R ◦ IdX = R.$ How do I go about doing it?
2
votes
2answers
46 views

Proving $\varphi(x^n)=\varphi(x)^n$ is a homomorphism for all $n\in\mathbb{N}$.

I began my proof using induction however I'm a little stuck on understanding what to for my inductive step. I'm looking for proof validation and tips to improve my proof-writing. Pf: Let $G$ and $H$ ...