# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 $a, b: \mathbb{N}\to\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty\} \cup \{\infty\}$ ...
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### 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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### 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 ...
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### 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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### 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?
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 ...
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 & ...