If you already have a proof for some result, but want to ask for a different proof (using different methods).

learn more… | top users | synonyms

0
votes
1answer
28 views

Proving sequence converges

I am trying to prove: Suppose that {an} and {bn} are two sequences such that {an} and {an + bn} converge. Prove that {bn} converges. Here is my first attempt: Proof: Suppose that {$a_n$} and {$b_n$} ...
0
votes
3answers
54 views

Prove if $f$ is continuous at a and $g$ is discontinuous at a, then $f + g$ is discontinuous at a

Hello I want to prove: if $f$ is continuous at $a$ and $g$ is discontinuous at $a$, then $f + g$ is discontinuous at $a$. But with the $\epsilon - \delta$ definition of continuity and discontinuity (I ...
3
votes
0answers
77 views

Most complicated proof of Pythagoras

Usually a mathematician aims for clarity and elegance when conducting a proof. However, the antimathematician buries all hope of assimilating intuition and reasoning. To illustrate this, I seek the ...
0
votes
1answer
10 views

Exact differential equation $f(t,x)+g(t,x)\dot x=0$ and constants

The differential equation $$f(t,x)+g(t,x)\dot x=0\tag{*}$$ with $(t,x)\in U\subset\mathbb R^2$ and $U$ open is called exact, if there is a continuous and differentiable function $F\colon ...
2
votes
2answers
53 views

(Correct Proof?) Show that every convergent sequence $(\mathbf{x}_{k})$ in $\mathbb{R}^{n}$ is bounded.

Now, this is a pretty simple proof and I just wanted some more experienced members here to have a look at it and maybe give me feedback on my proof idea for the statement in the title. I also found ...
8
votes
1answer
55 views

Evaluating a certain integral without the fundamental theorem

I'm a TA for a calculus course. And they recently began calculating definite integrals using a definition equivalent to Riemann's criterion. Of course, the type of things they were calculating were ...
2
votes
2answers
53 views

Find $\int_{-1}^{1} \frac{\sqrt{4-x^2}}{3+x}dx$

I came across the integral $$\int_{-1}^{1} \frac{\sqrt{4-x^2}}{3+x}dx$$ in a calculus textbook. At this point in the book, only u-substitutions were covered, which brings me to think that there is a ...
5
votes
1answer
100 views

Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers.

Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers. I can explain the answer but would like help translating it into ...
8
votes
5answers
129 views

Finding $\int \frac{dx}{a+b \cos x}$ without Weierstrass substitution.

I saw somewhere on Math Stack that there was a way of finding integrals in the form $$\int \frac{dx}{a+b \cos x}$$ without using Weierstrass substitution, which is the usual technique. When $a,b=1$ ...
2
votes
1answer
58 views

Can I assume that random variables with exponential distribution are positive?

Let $(Y_n)$ be i.i.d random variables following exponential distribution with parameter $1$. Let $X_n=\min(Y_1,\dotsc, Y_n)$ Prove that $ X_n \xrightarrow{P} 0$ It's easy to prove that ...
1
vote
2answers
41 views

Number of graph vertices of odd degree is even

This elementary result is normally stated as a corollary to the Handshaking Lemma, which says nothing about it other than that it's true. I wonder if there is more depth to this fact, in particular if ...
3
votes
0answers
73 views

Every skew-symmetric matrix has even rank [duplicate]

Let $F$ be a field where $char(F)\neq2$ and let $A$ be a skew-symmetric matrix over $F$. Prove that rank of $A$ is even. I think the best way to prove it, is using induction on size of $A$. for ...
1
vote
2answers
60 views

How to evaluate $\int \frac{dx}{(1-x)\sqrt{1-x^2}}$ without a trig substitution/parts?

I'd like to find $$\int \frac{dx}{(1-x)\sqrt{1-x^2}}$$ but without using a trig substitution or integration by parts. I can already see that $x=\sin \theta$ works it out quite nicely, but I was ...
6
votes
2answers
84 views

Find all $n$ such that $\sqrt{5n+2}$ is an integer.

Here is my solution. There is no such $n$. If $n$ is odd, then, then $5n+2 \equiv 7 \pmod {10}$. Else, $5n+2 \equiv 2\pmod {10}$. But, the quadratic residues of $10$ are only $0,1,4,9,6,5$. ...
1
vote
1answer
26 views

Proofing de Movire without Induction and in a neat way

The "usual way" gone for proving de Movire is via the road of induction. However this road get tiresome and thus wondered, if there were another way. However I came up with a proof that relies on ...
1
vote
1answer
28 views

Show that $\mathbb{Q}(\sqrt{2})$ is a field.

Proof: Since $\mathbb{Q}$ is a field, then $\mathbb{Q}$ is a domain. (Theorem: if $R$ is a domain, then $R[x]$ is a field.) By the theorem, $\mathbb{Q}[x]$ is a field. So, letting $x = \sqrt{2}$, ...
2
votes
2answers
59 views

On $\sigma-$ finite space $fg\in L^1$ for every $g\in L^q$ prove $f\in L^p$

Let $(X,\mathcal{A}, \mu)$ be an measure space. Let $f$ be an extended complex-valued $\mathcal{A}-$measurable function on $X$ such that $|f|<\infty$ $\mu$-a.e. on $X$. Suppose that $fg\in ...
3
votes
0answers
65 views

$fg\in L^1$ for every $g\in L^1$ prove $f\in L^{\infty}$

Let $(X,\mathcal{A}, \mu)$ be an arbitrary measure space. Let $f$ be an extended complex-valued $\mathcal{A}-$measurable function on $X$ such that $|f|<\infty$ $\mu$-a.e. on $X$. Suppose that ...
0
votes
0answers
27 views

Theoretical Math Sequence Proof

Suppose that {xn} is a sequences such that every subsequence {xni} has a subsequence {xnmi} that converges to x. Show that {xn} is bounded. I tried to do a proof by contradiction but am not sure if ...
2
votes
1answer
72 views

Prove that if $A$ and $B$ are square matrices and $AB$ is invertible, then both $A$ and $B$ are invertible

I already know how to prove this using the definition of inverse and the associative property of matrix multiplication, but I was wondering if this would also be a valid proof. As $A$ and$ B$ are $n ...
0
votes
1answer
43 views

Proving that a set has no largest member

Here's the question in mind. Let $$A = \left\{r : r \quad \text{is a rational number and} \quad r^2 < 2\right\}$$Prove that $A$ has no largest number. (Hint: if $r^2 < 2$, and $r > 0$, ...
0
votes
2answers
81 views

Proof that the difference of two positively squared integers never equals 1

This type of question is usually solved by a proof by contradiction, however I believe I have a direct proof of it, and I would like to know if its correct. Problem : Prove that there does not exist ...
1
vote
1answer
18 views

Prove that $|x-a_1|+|x-a_2|+|x-a_3|\geq a_3 - a_1$, for $a_1<a_2<a_3$, and determine the condition for equality.

I got this question from the first chapter of Courant and John's Introduction to Calculus and Analysis I. The problem is as follows: Prove that $|x-a_1|+|x-a_2|+|x-a_3|\geq a_3 - a_1$, for ...
1
vote
1answer
36 views

Primes Between $n$ and $2n$ For $n\ge6$

For $n\ge6$, there are at least two primes in the interval between $n$ and $2n$. Does anyone know of an already established and accepted proof for this? A reference would be helpful. I have read in ...
4
votes
0answers
84 views

Using the IVP definition of $\cos$ and $\sin$, how can we show that $\cos^2(x)+\sin^2(x) = 1$ without any “magic”?

One way to define the exponential, hyperbolic and circular functions is to assert that they're the unique solutions to certain IVP systems: The exponential function: $$\exp'(x) = \exp(x), \qquad ...
0
votes
1answer
128 views

Different proofs for $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ [closed]

Different proofs that show $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ cannot be the square of an integer, where n is a natural number.
1
vote
2answers
45 views

Alternate Proof to $f(e_G)=e_H$

Is this proof correct? Proof: Let for all $a$ and $e_G \in G$ we know that if $f$ is a homomorphism from ${(G,*)}$ to $(H,o)$ then, $f(a*e_G)=f(a)=f(a)$ o $f(e_G)$. Similairly ...
1
vote
0answers
105 views

Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions.

Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. I believe I have proved it for the case when both $y$ and $z$ are of the same parity. Case 1: When $y$ and $z$ are of the ...
2
votes
5answers
65 views

Prove $\sinh x > x$ for all $x >0$

I did a proof for $\sinh x > x$ for all $x > 0$. But I am not sure if the proof is mathematically valid. I started by showing that $\frac{d}{dx} \sinh x = \cosh x$ and that the limit of $\cosh ...
1
vote
0answers
13 views

alternative proof that a function is homogeneous of degree one

Given a (profit) function of the form $$ \pi(p) = \sup \{p.y:y \in Y\} $$, where $p \in R_+^k$ is a positive (price) vector and $Y \in R^k$ is a (production-possibility) set. I need to proof that ...
1
vote
1answer
33 views

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$. Now let $z$ be the point $l \cap m$. Let $n$ be the ...
0
votes
1answer
54 views

$10$-digit perfect squares that contain each of the digits $1, 2, 3, 4, 5$ twice

Are there any 10-digit perfect squares that contain each of the digits $1, 2, 3, 4, 5$ twice? Perfect squares belong to the set $\{0, 1, 4, 7\}$ modulo $9$ and any such number will be equal to ...
3
votes
7answers
106 views

Proof that $e^{-x} \ge 1-x$

My aim is to prove that $e^{-x} \geq 1-x$ for any $x \geq 0$. What I found so far is Bernoulli's inequality, which states that $$1+x\leq\left(1+\frac{x}{n}\right)^n\xrightarrow [n\to\infty]{} e^x$$ ...
4
votes
3answers
100 views

Alternate Proof for the Cancellation Laws

I will first state the theorem: Given a group $(G,*)$ the following laws apply for $a, b, c \in S $ If $a*b=a*c$ then $b=c$ If $b*a=c*a$ then $b=c$ Attempt at alternate Proof: Consider the ...
1
vote
2answers
75 views

Possibly not an acceptable proof for uncountablity of countable product of countable sets

Here is a text from the book Topology by Munkres: Theorem 7.7. $ \ \ \ $ Let $X$ denote the two element set $\{0,1\}.$ Then the set $X^\omega$ is uncountable. Proof. $\ \ \ \ $ We show that, ...
4
votes
2answers
63 views

Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$

Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$. What are the possible values of $$\frac{x^2+y^2-1}{xy}$$? I have discovered ...
2
votes
0answers
54 views

Simplifying Fraction for Nested Radicals

A while back, a problem asked me to simplify ...
5
votes
0answers
90 views

Proof in Algebraic Topology without appeal to intuition

My question arose from the proof of proposition 1.26 in Hatchers Algebraic Topology. There he constructs a space $Z$ from a path-connected space $X$ as follows: attach a set of 2-cells $e_\alpha$ ...
1
vote
4answers
57 views

If $f,g: X\to Y$ be two functions continuous on $X$ then show that $\{x:f(x)=g(x)\}$ is closed in $X$

Problem. Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be two topological spaces and $f,g:X\to Y$ such that $f$ and $g$ both are continuous on $X$. Show that the set $E:=\{x:f(x)=g(x)\}$ is ...
4
votes
1answer
47 views

Show $\dbinom{a}{b}\dbinom{b}{c}=\dbinom{a}{a-c}\dbinom{a-c}{a-b}$

I am trying to give a non-algebraic proof for this equality: $$\dbinom{a}{b}\dbinom{b}{c}=\dbinom{a}{a-c}\dbinom{a-c}{a-b}$$ So far, I could only use the identity $\dbinom{x}{y}=\dbinom{x}{x-y}$. ...
8
votes
1answer
140 views

Determinant of $n\times n$ matrix with parameter

Problem: Let $\delta \in \mathbb{R}^+$ and $n\in \mathbb{N}$. The matrix $A_n = (a_{i,j}) \in \mathbb{R}^{n\times n}$ is defined as $$ a_{i,j} = \prod_{k=0}^{i-2}\left((j-1)\delta +n-k\right) ...
4
votes
0answers
82 views

Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$ \int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $ e^{- x \log x} $ and ...
4
votes
1answer
69 views

How to prove this complex inequality elegantly?

Question Let $z_{1,2}\in U(0,1)\subset \Bbb C$, prove that $$\frac{|z_1|-|z_2|}{1-|z_1||z_2|}\le\left|\frac{z_1+z_2}{1+\overline{z_1}z_2}\right|\le\frac{|z_1|+|z_2|}{1+|z_1||z_2|}$$ Actually I ...
0
votes
1answer
16 views

Improving the proof for: $A$ is equicontinuous, show that $\overline A$ is equicontinuous

Let $A \subset C^0([a,b],\mathbb{R})$ Show that if $A$ is equicontinuous, then $\overline A$ is equicontinuous. Preliminary Proof: Let $(f_n) \subset \overline A$, let $\epsilon >0$ be given, ...
0
votes
0answers
36 views

Surface Area and Volume of a Torus Using Polar Coordinates

Can the volume and surface area of a torus be derived using double integrals and a coordinate transformation to polar coordinates where $x = rcos(\theta)$ and $y = rsin(\theta)$? Equation for the ...
1
vote
1answer
42 views

Reference request for “Elementary” Proofs of Picard's Great Theorem

This is Picard's Great Theorem; $\textbf{Great Picard Theorem.}$ Suppose an analytic function $f$ has an essential singularity at $z=a$. Then in each neighbourhood of $a$, $f$ assumes each complex ...
2
votes
1answer
32 views

proof - $\gcd(a, m) = \gcd(b, m) = 1 \implies \gcd(ab, m^2) =1$

I have a proof and want to know if its correct. Prove that $\gcd(a, m) = \gcd(b, m) = 1 \implies \gcd(ab, m^2) =1$ Proof: $ax_0 + my_0 = 1$ and $bx_1 + my_1 = 1$ $ax_0 = 1 - my_0$ and $bx_1 = 1- ...
0
votes
1answer
37 views

Proving that $z^4-6z^2+4z-3 = y^2$ has only one integer solution

I'm trying to prove the following result. Conjecture. If $z$ is an integer, and $z^4-6z^2+4z-3$ is a square, then $z=3$. A quick check modulo $9$ shows that $z=9w+3$ for some integer $w$. So for ...
2
votes
2answers
82 views

square matrix $\mathbf{A}$ with $\mathbf{A}^\intercal = -\mathbf{A}$, proof $\mathbf{A}$ is not invertible.

I tried proving that given a square $\mathbf{A}$ over $\mathbb{R}$ so that $\mathbf{A}^\intercal = -\mathbf{A}$, $\mathbf{A}$ is not invertible. I know that because the matrix is real and its ...
1
vote
0answers
8 views

How to prove mathematically these two different definitions of background-position property as equivalent?

I've been reading about how percentage values work for background-position position property. The official definition is that ...