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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11 views

How do I prove a sequence is disjunctive?

I wrote a random number generator with an unbounded state size. I don't know where to begin proving it to be (or proving it isn't) disjunctive. What would be a property of a disjunctive sequence, ...
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1answer
36 views

Computing the Lebesgue Integral of $\int_0^1 \frac{1}{\sqrt{x}}\; d\mu$

I am trying to compute the Lebesgue integral of $\int_0^1 \frac{1}{\sqrt{x}}\; d\mu$. I know that if a function $f$ is bounded on some set $X$ and is continuous almost everywhere on $X$, then the ...
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23 views

Prove that a constraints system is a manifold.

What shown to follow is a theorem taken from the text Analysis on Manifolds written by James Munkres. So at the page 11 of this document, if you like you can read the proof of the mentioned theorem. ...
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19 views

How to prove $A\subseteq B \wedge C:=\lbrace x:x\in A\wedge x\in B\rbrace \longrightarrow C=A$

How may I prove that such a statement if set operations are yet to be defined in my course (Introductory Proof-Writing). My questions: $1$. Is my proof alright? $2$ May you give any alternative proofs ...
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2answers
31 views

Proof by contrapositive for the statement $P \wedge Q \wedge R \ \Rightarrow S$

I am asking the question not purely for a logic exercise but I am just trying to prove something by contrapositive and this got me confused. So when $A$ is a statement where, say, three conditions ...
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1answer
35 views

How do I formulate a proof by equivalences in english?

I have a proof of the form Theorem. $A \iff \forall x D$. Proof. \begin{align} A &\iff \forall x B \\ & \iff \forall x C \\ & \iff \dots \\ & \iff \forall x D \end{align} QED. Note ...
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14 views

Prove differential equations $S(0)= S_0 > 0, I(0) = I_0 > 0, R(0) = R_0 \geq 0$ are monotonically decreasing

To model an infectious disease, we look at the following epidemiological sizes: S: "susceptible" - Amount of susceptible persons I: "infectious" - Amount of infected persons R: &...
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25 views

Proving that for language $L$, $L^*L = L^*$

The Question Prove that $L^*L = L^*$ for language $L$. My solution I know the standard way is to show $LHS \subseteq RHS$ and vice versa. I am wondering if my approach works though, and if not, why ...
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2answers
45 views

Proof by induction for all positive $n$

I have to prove by induction that the following inequality holds for all positive $n$. I am unsure on how to approach this problem so any hint would be immensely appreciated \begin{align}\binom{2n}{n} ...
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Proof: Graph Coloring algorithm needs at most twice as many colors to color a graph as the optimal solution

Given an algorithm (1) create the complement $\bar{G}$ of input graph $G$ (2) calculate a maximum matching $M$ on $\bar{G}$ (3) color the two vertices of every edge $e_i \in M$ with color $i$ (4) ...
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36 views

Can someone help me understand how the second FTOC works?

Can someone help me understand how the second Fundamental Theorem of Calculus works? If $F(x)$ is well-defined on the interval $[a,b]$ and $F'(x) = f(x)$. $$\int_{a}^{b}f(x)\,dx = F(b) - F(a).$$ I ...
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1answer
35 views

Show that the solution is asymptotically stable

Let $A$ be a constant $n\times n$-matrix and $u(t) \in\mathbb{R}^n$ be a continuous function defined on $\mathbb{R}$. Assume that the real part of any eigenvalue of A is negative. Show that solutions ...
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3answers
108 views

Proving $19 \mid 2^{2^n} + 3^{2^n} + 5^{2^n}$

Theorem. $19 \mid 2^{2^n} + 3^{2^n} + 5^{2^n}$, for all positive integers $n$. I'm tasked with proving the given theorem by induction. Here's where I've gotten so far... Proof. Clearly, the theorem ...
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2answers
40 views

How do I find the error within this Fibonacci Sequence proof that is trying to prove that f(5) = 4?

I am working on a problem in my textbook where I am given this proof dealing with Fibonacci numbers. The function $f$ is defined by $f(0) = f(1) = 1$ and for all $n\geq 2$, and $f(n) = f(n-1) + f(n-2)$...
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1answer
22 views

Propositional Logic: proof involving conditional statements and disjunction conclusion

Please show me how I should work on the following proof: https://imgur.com/a/lwBFVzY $${1.~~B\supset{\sim}(A\supset C)\\2.~~{\sim}C\supset{\sim}A\qquad/\therefore D\lor {\sim}B}$$ All I can think of ...
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1answer
27 views

Prove that $\boldsymbol{v}^T\boldsymbol{A}^{-1}\boldsymbol{v}\boldsymbol{z}^T\boldsymbol{A}\boldsymbol{z} \ge (\boldsymbol{v}^T\boldsymbol{z})^2$

I'm facing the following problem: Prove that $$ \boldsymbol{v}^\top\boldsymbol{A}^{-1}\boldsymbol{v}\boldsymbol{z}^\top\boldsymbol{A}\boldsymbol{z} \ge (\boldsymbol{v}^\top\boldsymbol{z})^2 $$ where $\...
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0answers
36 views

Stability of null solution

I need some help, I have to prove the stability of the null solution of the following differential equation : $$\frac{d}{dt}\begin{pmatrix} x\\ y \end{pmatrix} =\begin{pmatrix} -\sin(x) & -y + \...
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2answers
52 views

Proving the squeeze theorem

I know there are posts out there asking about this, but I didn't want to look at other solutions before I feel like I've solved it on my own or at least gotten close. If I'm on the wrong track, any ...
2
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1answer
29 views

Proving Zorn's lemma implies axiom of choice (trouble showing every chain has an upper bound)

I am attempting to prove the fact that Zorn's lemma implies the axiom of choice, however my proof falls short when I try to prove every chain has an upper bound. I understand there are other proofs ...
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1answer
35 views

Using existential statements in informal mathematical reasoning

In symbolic logic, existence statements are normally written as $\exists x P$ where $P$ is some smaller statement. The $x$ is bound by the existential quantifier and the scope of $x$ is $P$. When ...
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2answers
139 views

Proving that $x(x^2-1)(x^2-10)=c$ cannot have five integer solutions for any real $c$

I found this question that caught my attention at MSE and I did a solution, but I suspect something is wrong with the solution. Original problem says: Prove that for any real values of $c$, the ...
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0answers
53 views

Regular branch and analytic function

Let $\phi(z)$ be the regular branch of analytic function $\sqrt{\frac{1+z}{1-z}}$ in domain C$ \backslash$[-1,1] fixed by condition $ϕ(0 + i0) = lim_{ε→0} ϕ(iε) = 1$.In my case I want to find ϕ(−i), ...
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1answer
38 views

Contradictory proofs of $ax+by\ge{ay+bx}$ when $a\ge{b}$ and $x\ge{y}$

Here is a problem I found recently: If $a\ge{b}$,$x\ge{y}$, prove that $ax+by\ge{ay+bx}$ for all real numbers $a,b,x,y$. I found $2$ solutions to the problem, which possibly contradict each other. ...
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4answers
110 views

Proof by going the reverse way

Proving a mathematical statement usually involves coming to the required result from a known/given result(s). Am I allowed to do this the other way round i.e. coming to a known result from the ...
2
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3answers
50 views

Divisibility of an expression by 11

How to prove that $5^{5n+1}+4^{5n+2}+3^{5n}$ where $n\in \mathbb{N}$ is divisible by $11$ using mathematical induction? I have tried and got to this $$5 \cdot 25 \cdot 25 \cdot 5^{5k+1}+4\cdot 16 \...
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2answers
418 views

Why don't mathematicians introduce intuition behind concepts as physicists do?

First of all please don't be angry - if anyone might be - and thoughtlessly downvote this post. I'll make it clear that I'm not here to criticise mathematicians - but rather to understand. I ...
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3answers
61 views

Prove $f'(0) = 0$ if $f$ is even

Let $f$ be an even function on the reals. Prove $f'(0)$ is either $0$ or undefined. Intuitively, $f(0) + h\cdot f'(0) \approx f(h) = f(-h) \approx f(0) - h \cdot f'(0) $ for small $h$. If $h \neq 0$,...
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1answer
46 views

Prove that a function is differentiable at $0$ when $h(x)=x^2f(x)$.

Let $f : (−1, 1) \to\mathbb R$ be a bounded function. Let $g : (−1, 1) \to\mathbb R$ defined by $g(x) = xf(x)$ is continuous at $x = 0$. Use this result to prove that the function $h : (−1, 1) \to\...
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1answer
27 views

Prove that $f*g$ is differentiable at $0$ without product rule.

Suppose that $f : R → R$ and $g : R → R$ are continuous functions satisfying (i) $f(0) = 0$, (ii) $f'(0)=3$, and (iii) $g(0) = 2$. Prove that $f*g$ is differentiable at $0$, and find $(f*g)'(0)$. Note:...
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2answers
49 views

Convergence of a Sequence Proof (Introduction)

Problem: Let $a_n=\frac{1}{n^2}$. It converges to $0$. Proof that it converges to $0$ Proof: Let $\epsilon>0$ be given. Let $N=\left \lceil \frac{1}{\sqrt{\epsilon}} \right \rceil$. For $n>N$ we ...
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2answers
35 views

Proving that the lower Riemann sum of this modification of Dirichlet function is $0$.

Consider the following variant of the Dirichlet function-- $f : [0, 1] \rightarrow (0, \infty)$ where $$f(x) = \begin{cases} \dfrac{1}{b}\; \text{ if $x$ is rational and $x= \dfrac{a}{b}$ ...
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1answer
32 views

Explain: If $\left \{ a_n \right \}_{n\in\mathbb{N}}$ converges with a limit $a$, then any subsequence converges with the same limit.

Lemma. If $\left \{ a_n \right \}_{n\in\mathbb{N}}$ converges with a limit $a$, then any subsequence converges with the same limit. Proof. Let $\left \{ a_{n_k} \right \}_{k\in\mathbb{N}}$ be a ...
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1answer
62 views

how to prove this calculus 2 result? [closed]

If $f(x)\ge0$ and monotonic for every $a<x<\infty$, and $$\int_a^\infty |f(x)\sin(x)|\,\mathrm dx$$ exists, then $$\int_a^\infty f(x)\,\mathrm dx$$ exists.
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1answer
30 views

Prove that $\mathcal{F}\cdot\mathcal{G}$ is the glb of the set $\{\mathcal{F},\mathcal{G}\}$

I want to refer to Exercise 26 part (c) of Section 4.6 of Velleman's 2nd Edition book. The exercise is as follows: Supose $A$ is a set. If $\mathcal{F}$ and $\mathcal{G}$ are partitions of $A$, then ...
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1answer
45 views

can anyone help me about the 'cauchy product rule' [closed]

can anyone help me about the 'cauchy product rule' part Property -4: (Addition Theorem) from generating function, $$E_n(x+y) = \sum_{k=0}^{n} \binom{n}{k}E_k(x)y^{n-k}$$ Proof: - We know from the ...
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0answers
23 views

Prove that set of all finite subsets of N with n elements is denumerable [duplicate]

I need to prove that the set S of all finite subsets of ℕ with exactly n elements (where n is a natural number) is denumerable. I think I need to set up an injection from S to ℕ since ℕ is a ...
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0answers
35 views

How can I proof that $\lim\limits_{n\to\infty} \dfrac{n^b}{a^n}= 0\;,\;$ when $\;a > 1,\; b > 0\;.$ [duplicate]

I am having a hard time with this problem: I tried by using the sandwich theorem but I failed. $\lim\limits_{n\to\infty} \dfrac{n^b}{a^n}= 0\;,\;$ when $\;a > 1,\; b > 0\;.$ Could anyone please ...
1
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1answer
46 views

Showing $(n+m)^{\underline{k}}=\sum_{v=0}^\infty{k\choose v}\cdot{(m)^{\underline{k-v}}}\cdot(n)^{\underline{v}}$ for falling factorials

Falling and rising factorials are defined as $$ \color{blue}{n^{\underline{k}}}=\,\,\color{blue}{\prod_{j=0}^{k-1}(n-j)} \qquad\qquad \color{blue}{n^{\overline{k}}}=\,\,\color{blue}{\prod_{j=0}^{k-1}(...
3
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1answer
51 views

Let a: N → R be a sequence with a (n) → A ∈ R as n → ∞. Find B ∈ R such that for b: N → R with b (n): =. $\frac{1}{n}\sum_k^n$.

I am having trouble with a problem for 3 days. I am having a problem understanding this one. To be honestly, I don't know the first step to take to solve it. I solve the other problems like this using ...
0
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1answer
42 views

Proving $4^n - 1 - 7n > 0$ for all $n \geq 2$ by induction [closed]

Prove that $4^n - 1 - 7n > 0$ for all $n \geq 2$ by induction. I am struggling with the part that involves $k+1$ but I am not able to beyond 3 steps and I get stuck with it.
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0answers
51 views

Prove if an Integer is Divisible by 4, then it can be Written as the Sum of Two Odd Integers.

So far, I keep showing myself the definitions of even and odd integers over and over but I do not know how to show this without examples. And, well... there are so many. To prove this, I have.. Taking ...
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0answers
30 views

Feedback to proof about limit for $a(n)= \frac{n!}{n^{n}}$ and $b(n):=\frac{3^{n+2}- 2^{n}}{3^{n} + n}$

I have the following task: Determine whether the following defined sequences $a, b: \mathbb{N}\to\mathbb{R}$ or $a, b: \mathbb{N}\to\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty\} \cup \{\infty\}$ ...
5
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2answers
221 views

Maximizing $x^2y$ given $x^2+y^2=100$, without using the AM-GM inequality and calculus tools

Problem says: Let $x^2+y^2=100$, where $x,y>0$. For which ratio of $x$ to $y$, the value of $x^2y$ will be maximum? I know these possible tools: AM-GM inequality Calculus tools Here, I want to ...
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0answers
28 views

Let $a > 1, b > 0$. Prove that $\lim_{n\to \infty} \frac{n^{b}}{a^{n}}=0 $ and $ \lim_{n\to \infty} \sqrt[n]{b} = 1$

I run out of ideas (after three days) of trying to solve this problem: Let $a > 1, b > 0$. Prove that $\lim_{n \to \infty} \frac{n^{b}}{a^{n}}=0 $ and $ \lim_{n \to \infty} \sqrt[n]{b} = 1$ I ...
0
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2answers
69 views

How to show, that $z(y+1)=4(y+1)$ is $z=4$ rigorously?

For context: Theorem. There is a unique real number $x$ such that for every real number $y$, $xy+x-4=4y$. Proof. We show the existence by choosing $x=4$. Substituting it in gives. $xy+x-4=4y+4-4=4y$. ...
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2answers
36 views

Steps for how to Prove that for any a,b in R with a<b one has that. [closed]

I am in an intro to higher math class and the book we were given doesnt have any inequality proofs similar to this. Any help in the right direction would be helpful. Prove that for any $a,b \in \...
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2answers
48 views

Prove the following statements. (a) $\|fg\|_u \le \|f\|_u\|g\|_u$; (b) $\sup_{n\in N} \|f_n\|_u < \infty$

Suppose that $f_n, g_n, f, g:\mathbb R^d→\mathbb R$ are all bounded functions, and suppose further that $f_n→f$ uniformly and $g_n→g$ uniformly. Prove the following statements. (a) $\|fg\|_u \le \|f\|...
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0answers
39 views

Prove the theorem $(P\to Q)\to R$ [closed]

How do you prove the theorem $(P\to Q)\to R$? Is there more than one proof method that can be used?
0
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1answer
14 views

Composition of two functions - Taylor series

Let $f$ and $g$ be $n$-times differentiable functions, and let us assume that the composition $ F (x) = f (g (x)) $ is well-defined on an interval. Let us say that the composition is also $n$ times ...
1
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2answers
29 views

Diagonalization of a particular matrix

I want to prove the following fact: let $A$ be a non-zero square matrix (matrix of an endomorphism in some basis) whose column vectors are the same, i.e. $A = \begin{pmatrix} a_1 & \dots & ...

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