Proofs

Let f be differentiable on R and suppose that there exists M > 0 such that, for any x, y $\in$ R, $|f(x) - f(y)| \leq M|x-y|^2$. Prove that f is a constant function.

I’m pretty new to proof—based mathematics and so far I’ve noticed that one type of proof I seem to struggle the most with is proving an existence claim. Particularly, showing the existence of some obj...

Need help for introductory level discrete math course, proof by induction only. See attachment for details

I need to prove this with the triangle inequality together with a case distinction.If possible, please explain what you do every step and why this is legal. Thank you very much <3

Hey guys, Maybe someone of you can help. I need to prove the following through inclusion: (A∆B) ∪ A = A ∪ BAny suggestion on how to solve this?

1st-year undergraduate combinatorics. 

Trying to prove that the statement is true, which I am sure it is. Much like the last problem posted here, I am unsure how to go about solving itI do not believe a negation is necessary but I could be...

I am already sure that the original statement in the image is false, however, it needs to be proven by showing that the negation is trueBecause of the domain, only integers can be used if plugged in

Let $\Sigma$ be a Polish space and $M_1(\Sigma)$ be the space of probability measures in $\Sigma$. Prove that\left\lVert\nu-\mu\right\rVert_{var}^2\leq 2\textbf{H}(\nu|\mu), \ \ \ \ \mu,\nu\in\textbf{...

Critique my proof on correctness, structure, etc. Theorem. Suppose $A$ and $B$ are sets. Then $A \times B = B \times A \iff A = \emptyset, B = \emptyset,$ or $A = B$Proof. ($\rightarrow$) Assume for t...

Critique my proof on correctness, structure, etc.Theorem. For every integer n, $6 \vert n$ $\ \text iff$ $2 \vert n$ and $3 \vert n$. Proof.  Let $n$ be an arbitrary integer.$(\rightarrow)$ Suppose $6...

Fix any errors in correctness, structure, etc.Theorem. Suppose $B$ is a set and $F$ is a family of sets. If  $\bigcup F \subseteq B$, then $F \subseteq \wp(B)$.Proof. Suppose $\bigcup F \subseteq B$....

Last time I did this I wasn't aware how difficult the proof was, but this one should be fairly straightforward. I upped the bounty regardless, though.Critique my proof of the following theorem on corr...

Fix any errors in correctness, structure, etc.Theorem. Suppose that a and b are nonzero real numbers. Prove that if $a < 1/a < b < 1/b$, then $a < −1$. Proof. By contrapositive, if  $a \ge...

I'm going through a book on mathematical proof and I'm really confused on what this page means. Specifically, what is i and I and what is the relationship between them?Thanks.Page:https://ibb.co/QvRGF...

There is a puzzle with 8 pieces and one single board to put them.The board is labeled with months and dates.Your goal is to place this 8 pieces in a way that leaves only 2 squares open.That 2 square d...

Use induction to prove that for any natural n the following holds (see title)

Let $f\in C (\mathbb{R})$ and $f_n=\frac{1}{n}\sum\limits_{k=0}^{n-1} f(x+\frac{k}{n})$. Prove that $f_n$ converges uniformly on every finite interval.

Recall the following definition of a Poisson process with rate λ > 0 : a counting process {N(t)} with N(0) = 0 is called a Poisson process with rate λ if the following are true: it has independent...

Let G be a finite group whose order is not divisible by $3$. Suppose that $(ab)^3 = a^3 b^3$ for all $a, b \in G$. Prove that G must be abelian.

https://imgur.com/a/ZvsZ7q6If someone can help me with the two proofs in the Imgur link, it would be very helpful. Thanks in advance!

prove