Articles of proof explanation

is it correct or not?

let $u=u(x,t)$ and $$w(z,\tau)=\frac{e^{\tau}}{\sqrt2}u(\sqrt2 e^{-\tau} (z-\beta)+\beta,T-e^{-2\tau})$$ is it correct that $u$ satisfies $$\partial_t u-\frac{\partial^2_x u}{1+(\partial_x u)^2}+\frac{n-1}{u}=0$$ if and only if $w$ satisfies $$\partial_{\tau} w-\frac{\partial^2_z w}{1+(\partial_z w)^2}+z\partial_zw-w+\frac{n-1}{w}=0$$ i have tried over and over and i did not find the equivalence

I did not understand one thing in the proof of substitution lemma?

The substitution lemma in lambda-calculus is proved by the following way, but I just did not understand the application of induction hypothesis in it. The lemma as shown below, where x and y are distinct and x is not among the free variables of L: M[x:=N][y:=L] equals M[y:=L][x:=N[y:=L]] to prove that in the case where […]

Show that $\left | \sin a – \sin b \right | \leq \left | a-b \right |$

Show that $$\left | \sin a – \sin b \right | \leq \left | a-b \right |$$ I saw a proof using The Mean Value Theorem, but I could not grasp it well. Are there any other proofs or can someone clarify it by using The Mean Value Theorem?

Show that: $\lim\limits_{n\to\infty} \sqrtn = 1$

First of all, I already know the common proof for this limit. My question concerns a specific proof that I could not deal with. It starts with defining a sequence $\{x_n\}_{n=1}^{\infty}$ with the general term $x_n=\sqrt[n]{n}-1$ and then shows that this sequence converges to $0$. We have $n=(1+x_n)^n\geq \frac{n(n-1)}{2}x_n^2$ Up to this point everything is […]

Proving an inequality

The sequence of real numbers $\ a_1,a_2,a_3…..$ is such that $\ a_1=1$ and $$a_{n+1} = \left(a_n+\frac{1}{a_n}\right)^{\!\lambda} $$ where $\ \lambda >1$ Prove by mathematical induction that for $n\geq 2$ $$a_n\geq2^{g(n)} $$ where $g(n) = \lambda^{n-1} $ Solved Prove also that for $\ n\geq 2$, $$ \rm \large \frac{a_{n+1}}{a_n} > 2^{(\!\lambda-1)g(n)}$$ Attempt $$ \rm \large \frac{a_{n+1}}{a_n}-2^{(\!\lambda-1)g(n)}=\frac{a_{n+1}}{a_n}-2^{(\!\lambda-1)g(n)}$$ […]

Ground plan of Backward direction (<=) – Let $p$ be an odd prime. Prove $x^{2} \equiv -1 \; (mod \, p)$ has a solution $\iff p\equiv 1 \; (mod 4)$

Apply the identity $p-i \equiv -i \mod p$ for $i=1, \ldots$ to the pink factors $ \begin{align} \color{seagreen}{ (p-1)! } = 1\times 2\times\cdots\times \dfrac{p-1}{2} & \times \quad \color{magenta}{\dfrac{p+1}{2} \times\cdots\times(p-2)\times(p-1)}\ \\ = \quad \dfrac{p-1}{2}! \quad & \times \quad \color{magenta}{ \dfrac{p-1}{2} \times\cdots\times(2)\times(1) \times \quad (-1)^{ \dfrac{p-1}{2}} } \\ & \equiv \color{seagreen}{ ((\frac{p-1}{2})!)^{2} \cdot (-1)^{ \frac{p-1}{2} } } […]

Derivation of Variance of Discrete Uniform Distribution over custom interval

I’m trying to prove that the variance of a discrete uniform distribution is equal to $\cfrac{(b-a+1)^2-1}{12}$. I’ve looked at other proofs, and it makes sense to me that in the case where the distribution starts at 1 and goes to n, the variance is equal to $\cfrac{(n)^2-1}{12}$. I want to find the variance of $unif(a,b)$, […]

Prove that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $

I need help with this exercise from the book What is mathematics? An Elementary Approach to Ideas and Methods. Basically I need to proove: $$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $$ $i)$ Particular cases $ Q(1) = \frac{1}{2} ✓ $ ———- $ P(1) = 2 – \frac{1+2}{2^1} = \frac{1}{2} ✓ $ $+ = 1 […]

Definition of hyperbolic lenght.

Theorem 1: Let $\text{arc(AB)}$ be an arc of an equidistant curve (Which can be a circle, a horocircle or an equidistant line) and $(A^{n})$ a sequence of partitions of the arc $\text{arc(AB)}$ such that $\lVert A^{n}\rVert \rightarrow 0$ and $l(A^{n}) \rightarrow l$, where $l>0$. Then the lenght of the arc $\text{arc(AB)}$ is $l$. Obs.: Norm […]

Logic Behind Epsilon-Delta Proofs (Single-Variable Calculus)

Most of what I am asking is based off this (fairly popular) article I’ve read here :, but most lecturers, use this same process to tackle epsilon-delta proofs, so what I am asking should be pretty universal to epsilon delta proofs. Epsilon-Delta Definition of a Limit I’ve just referenced this here, for some added […]