Articles of alternative proof

Proving graph connectedness given the minimum degree of all vertices

I know that this is a repeat of a previous question asked with a similar title, but I didn’t want to revive an old thread. The solution presented in that thread seems to be the common one, but I was wondering if the alternate I came up with holds up and satisfies the claim. The […]

Proof that negative binomial distribution is a distribution function?

In my textbook, a clear proof that the Geometric Distribution is a distribution function is given, namely $$\sum_{n=1}^{\infty} \Pr(X=n)=p\sum_{n=1}^{\infty} (1-p)^{n-1} = \frac{p}{1-(1-p))}=1.$$ Then the textbook introduces the Negative Binomial Distribution; it gives a fairly clear explanation for why the PMF of a Negative Binomial random variable $N$ with parameter $r$ is $$p\binom{n-1}{r-1}p^{r-1}(1-p)^{n-r} = \binom{n-1}{r-1}p^{r}(1-p)^{n-r} $$ […]

A combinatorial proof of $\forall n\in\mathbb{N},\,\binom{n}{2}=\frac{n(n-1)}{2}$

The property $\forall n\in\mathbb N,\,\binom{n}{2}=\frac{n(n-1)}{2}$ was given in our first chapter on probability theory among binomial coefficients’ properties. It is really easy to prove with the formula $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, yet it is mentioned in other course notes, so I thought it has other combinatorial meanings and that it may even be quite special (like the fact […]

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at, They say that since $H\cap(C+K)= K$, we have $H\cap C=\{e\}$. But how do they have that this part $H\cap(C+K)= K$? Can someone […]

Continuity of polynomials of two variables

I’m working on a proof of the following statement (from T.Tao’s Analysis 2 book): “Let $n,m\geq 0$ be integers and suppose that for every $0\leq i\leq n$ we have a real number $c_{ij}$. Form the function $P\colon\mathbb{R}^2\to\mathbb{R}, P(x,y):=\sum_{i=0}^n\sum_{j=0}^m c_{ij}x^i y^j$. (1) Show that $P$ is continuous. (DONE) (2) Conclude that if $f\colon X\to\mathbb{R}$ and $g\colon […]

The center of a group with order $p^2$ is not trivial

Let $p$ be a prime and $G$ be a group of order $p^2$. Show that $Z(G)\neq 1$. Is there a proof of this nice fact that doesn’t use the class equation?

Measure theoretic proof of $|\Bbb{Z}^d/A\Bbb{Z}^d| = |\det(A)|$

Let $A \in \Bbb{Z}^{d\times d}$ be an invertible matrix with entries in $\Bbb{Z}$. It is well-known (and can be proved using algebraic properties of matrices) that the index of the group $A \Bbb{Z}^d \leq \Bbb{Z}^d$ in $\Bbb{Z}^d$ is given by $$|\Bbb{Z}^d/A\Bbb{Z}^d| = |\det(A)|. \qquad (\dagger)$$ Looking at the right hand side of this question, I […]

Prove withoui calculus: the integral of 1/x is logarithmic

It was known in the 17th century that the function $$ t \mapsto \int_{1}^{t} \frac{dx}{x} $$ is logarithmic: a geometric sequence in the domain produces an arithmetic sequence in the codomain. This is. of course, easy to prove with the fundamental theorem of calculus. But is there a simpler, perhaps geometric, way of proving this?

On the existentence of an element of a group whose order is the LCM of orders of given two elements which are commutaive

I came up with the following proposition. Proposition Let $G$ be a group. Let $x, y$ be elements of finite order in $G$ such that $xy = yx$. Let $n$ be the order of $x$. Let $m$ be the order of $y$. Let $l =$ lcm$(n, m)$. Then there exists an element of order $l$ […]

Make this visual derivative of sine more rigorous

Is this the correct way to make this visualization of the derivative of sine more… rigorous? At least, for $u\in(0,\pi/2)$. Borrowed from Proofs without words. To try to make this rigorous, I argued that when $u\pm\Delta u$ is in the first quadrant, that we have the following geometrically obtained bounds: $$\frac{\sin(u+\Delta u)-\sin(u)}{\Delta u}<\cos(u)<\frac{\sin(u-\Delta u)-\sin(u)}{-\Delta u}$$ […]