For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

learn more… | top users | synonyms

5
votes
4answers
172 views

prove $s(x+y)=s(x)s(y)$

I am asked to prove the following: Let $s(x):=\sum_{n=0}^{\infty}\binom{x}{n}$. Then $s(x+y)=s(x)s(y)$. I don't know how to start. I am thinking about $\exp(x)$ function with ...
3
votes
1answer
206 views

Any positive rational number can be expressed in one and only one way in the form …

I am attempting Miscellaneous Examples on Chapter 1, Number 2, from Hardy's Course of Pure Mathematics. Question Any positive rational number can be expressed in one and only one way in the form: ...
5
votes
1answer
116 views

The author of my book simplifies his solutions to an extent that I am uncomfortable with, so are my solutions to homework over doing it?

This question can be summarized as: How explicit does one need to be when writing proofs? To what extent can one implicity write a proof safely? The first chapter of our text in elementary discrete ...
4
votes
1answer
877 views

How do I show that two sets are equal.

This is an ever so slightly modified version of a question from my book. My teacher went over this with me, but I would like an explanation that I can keep coming back to until I have this method ...
36
votes
7answers
3k views

How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?

I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by ...
2
votes
1answer
145 views

How does this actually solve this simple proof?

I'm having a really hard time learning proofs, and cannot pick up on why this is actually proved. This isn't a homework problem, it's straight from the solutions manual (it just doesn't tell me why ...
8
votes
2answers
243 views

Why is the induction proof not sufficient? Topology…

I went to a exercise class and I got really confused. Consider the problem: Let $\lbrace A_{n} \rbrace$ be a sequence of connected subspaces of $X$, such that $A_{n}\cap A_{n+1}\neq \emptyset$ for ...
1
vote
1answer
97 views

proving compactness and convexity of a set

Suppose functions $f(x)$ and $g(x)$ are continuous with domain $X \subset \mathbb{R} $ which is nonempty, convex and compact, can we show that $$S \equiv (f(x), g(x)) $$ for all $x \in X$ is ...
3
votes
1answer
199 views

How should someone release their proof to the world? [closed]

Lets say someone (a reputable or non reputable mathematician) has come up with a remarkable one page proof to a famous maths problem. Lets say the proof is likely correct but hasn't been released to ...
1
vote
1answer
153 views

prove absolute convergence $\sum_{n=0}^{\infty}\binom{x}{n}$

i am trying to prove that this series $\sum_{n=0}^{\infty}\binom{x}{n}$ is absolute convergent. let $\binom{x}{n}:=\frac{x^n}{n!}$ but i am stuck on the way, can someone please help me out. my steps ...
13
votes
4answers
715 views

Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$ [duplicate]

Yesterday, my uncle asked me this question: Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$. How can we do this? Note that this is not a diophantine ...
1
vote
2answers
67 views

Does $f(s)\not= 0 \Rightarrow g(s)=0$ imply that $g(s)=0\Rightarrow f(s)\not= 0$?

I have the following implication: $$f(s)\not= 0\Rightarrow g(s)=0$$ Then we can deduce that its converse is also true. $$g(s)\not= 0\Rightarrow f(s)=0$$ where ...
2
votes
3answers
1k views

If both roots of the Quadratic Equation are similar then prove that

If both roots of the equation $(a-b)x^2+(b-c)x+(c-a)=0$ are equal, prove that $2a=b+c$. Things should be known: Roots of a Quadratic Equations can be identified by: The roots can be ...
-2
votes
2answers
195 views

Does complementary relation($\overline R$) is transitive?

Let $R$ be a relation that is transitive. Does complementary of $R$ ($\overline R$) is transitive?($\overline R$is hold transitive)
0
votes
0answers
32 views

comparing solutions

Given $f(x)$ and $g(y)$ with $$f'(x)<0, g'(y)<0$$ and $h(x)$ and $m(y)$ with $$h'(x)<0, m'(y)<0$$ Moreover, when $k>0, c>0$ are constants, I want to compare the solutions to ...
14
votes
7answers
885 views

$\infty - \infty = 0$ ?

I am given this sequence with square root. $a_n:=\sqrt{n+1000}-\sqrt{n}$. I have read that sequence converges to $0$, if $n \rightarrow \infty$. Then I said, well, it may be because $\sqrt{n}$ goes to ...
1
vote
2answers
282 views

Why is this last step even necessary in this proof with open sets?

Let $E^o$ denote the set of all interior point of a set $E$. Prove that $E^o$ is always open. Proof: For $p \in E^o$, there is a neighborhood $N_r (p) \subset E$. Since neighborhoods are open, for ...
5
votes
3answers
218 views

In a proof that is reliant on proven theorems, does one assume the reader's familiarity with said theorems, or explicitly include their logic?

In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic?
1
vote
2answers
164 views

Analyzing a sequence and continuity proof

I am trying to understand the following proof: Given $f$ is continuous, prove that for every convergent sequence $(x_n) \to a$ that $\lim_{k\to \infty}f(x_k) = f(a)$ So the prove goes like this ...
3
votes
0answers
945 views

Prove that the set Aut(G) of all automorphisms of the group G with the operation of taking the composition is a group

Let $G$ be a group. Say what it means for a map $\varphi: G \rightarrow G$ to be an automorphism. Show that the set-theoretic composition $\varphi \psi = \varphi \circ \psi$ of any two ...
2
votes
2answers
236 views

understanding the Golden ratio intuitively

i am very interested in Golden Ratio and its value. the Golden Ratio itself is not hard thing to visualize and understand in 5 minutes. But i am trying to reach the historical, logical reasons of ...
15
votes
3answers
627 views

prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit

i am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all ...
1
vote
1answer
116 views

Proof on countably discrete subset of any Hausdorff space.

I have this question for several days: Let $X$ be a topological space and $X$ is Hausdorff. $C$ is an countable discrete subset of $X$. Then does there exist a disjoint family of open sets $\{U_x: ...
3
votes
2answers
67 views

$a_{n}$ converges and $\frac{a_{n}}{n+1}$ too?

I have a sequence $a_{n}$ which converges to $a$, then I have another sequence which is based on $a_{n}$: $b_{n}:=\frac{a_{n}}{n+1}$, now I have to show that $b_{n}$ also converges to $a$. My steps: ...
3
votes
5answers
185 views

how do you prove this set problem?

I'm trying to teach myself set-theory. I have been unable to prove algebraically that: $(A \cup B) \cap \overline{(A \cap B)} = (A \cap \overline{B}) \cup (\overline{A} \cap B) $ I know it's ...
4
votes
2answers
204 views

Is this proof, that $\sqrt{n}$ is irrational for all non-square $n \in \mathbb{N}$, correct or not?

Prove that the square root of all non-square numbers $n \in \mathbb{N}$ is irrational I have made an attempt to prove this, I don't know if it's correct though: Take a non-square number $n \in ...
1
vote
2answers
63 views

Polynomial Rewriting Proof

Note. Please provide only a hint along with some explanation, but not the answer. I want to struggle with this problem. This is not homework. Show that for any number $c$, a polynomial $ P(x) = ...
1
vote
2answers
186 views

Prove $F_{n+1}F_{n-1}-(F_{n})^2=(-1)^n$ without induction

I am asked to pove the statement about fibonacci sequence. The task is from the passage about series and sequences. But the proof seems to need induction way, doesn't it? Prove the statement ...
2
votes
0answers
26 views

check for convergence $\frac{1}{2}(a_{n-1}+a_{n-2})$ [duplicate]

Possible Duplicate: Does $x_{n+2} = (x_{n+1} + x_{n})/2$ converge? i am asked to prove the convergence of this sequence. What given is, is this: $a_{0}:=a$, and $a_{1}:=b$ and for all ...
5
votes
4answers
572 views

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$

Prove that $\cot^2{(\pi/7)} + \cot^2{(2\pi/7)} + \cot^2{(3\pi/7)} = 5$ . I am sure this is derived from using roots of unity and Euler's complex number function, but I am very uncomfortable in these ...
2
votes
1answer
562 views

The closure of $S$ is the set of all limits of sequences of points in $S$ that converge in $E$

My book asks the reader to prove that the closure of $S$ (the intersection of all closed sets in $E$ that contain $S$) is the set of all limits of sequences of points in $S$ that converge in a metric ...
6
votes
4answers
153 views

prove $\lceil{x}\rceil=-\lfloor-x\rfloor$

i am trying to prove that $\lceil{x}\rceil=-\lfloor-x\rfloor$, but having difficulties to prove. the definitions are: $\lceil{x}\rceil:=m-1<x\leq m$ and $\lfloor{x}\rfloor:=n\leq x<n+1$. how ...
6
votes
5answers
196 views

how to prove $\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$

i am asked to prove this statement: $$\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$$ Now after several attempts, i am lost not knowing where and how to start. if I use induction, i am stuck on ...
4
votes
4answers
174 views

Help in proving $ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum\limits_{k=0}^{n} \frac{1}{k!} < 3 $.

I am trying to prove this statement for all $ n \geq 1 $ using induction: $$ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum_{k=0}^{n} \frac{1}{k!} < 3. $$ I said: Base case $ n = 1 $: $$ \left( ...
2
votes
1answer
728 views

Checking if my proof for path and walk in graph theory is correct

I am trying to prove that if there exists a walk in a graph from $v$ to $w$, then there exists a path in the graph from $v$ to $w$ where $v$ and $w$ are vertices of graph $g$... I am not sure if I ...
11
votes
2answers
180 views

prove $\binom{n}{k}\frac{1}{n^k}\leq\frac{1}{k!}$

i am learning maths so fast here in MSE, thank you guys so much for being here to help us! so now, my next step towards proficiency: :). i am trying to prove that ...
4
votes
1answer
170 views

prove $x \mapsto x^2$ is continuous

I am to show the continuity of this function with the help of $\epsilon$-$\delta$ argument. The function is: $g: \Bbb{R} \rightarrow \Bbb{R}$, $x \mapsto x^2$. Given the $\epsilon$-$\delta$ ...
2
votes
1answer
300 views

prove with $\epsilon$-$\delta$-argument: $x\rightarrow |-2x+3|$ is continuous

i am asked to prove with $\epsilon$-$\delta$-argument that $x\rightarrow |-2x+3|$ is continuous my steps: Definition of $\epsilon-\delta$-argument: $\forall \epsilon >0 \exists \delta>0$ with ...
2
votes
3answers
2k views

Prove that every bounded sequence in $\Bbb{R}$ has a convergent subsequence

I am to prove that every bounded sequence in $\Bbb{R}$ has a convergent subsequence. I am stuck not knowing how and where to start.
7
votes
8answers
1k views

Proof that $\sqrt{5}$ is irrational

In my textbook the following proof is given for the fact that $\sqrt{5}$ is irrational: $ x = \frac{p}{q}$ and $x^2 = 5$. We choose $p$ and $q$ so that the have no common factors, so we know that $p$ ...
4
votes
4answers
271 views

Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ ...
44
votes
3answers
7k views

Proof by contradiction vs Prove the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...
3
votes
3answers
121 views

Uniqueness of primitives

We all know that a primitive $F(x)$ of a function $f(x)$ is the function which derivative is $f(x)$. For example: $$\int 2x dx = x^2 + C$$ where $x^2 + C$ is the set of all primitives of the ...
1
vote
1answer
71 views

How can I improve this proof?

I am trying to write a proof that $a \vert b$ if and only if $da \vert db$. This is what I have so far: $da \vert db$ if and only if $\lfloor db / da \rfloor = db / da$ $\frac{db}{da} = ...
5
votes
1answer
295 views

Proof of pythagorean theorem

Any one seen this proof before? $$\frac{d}{dx} \sin(x)^2=2\cos(x)\sin(x)$$ $$\frac{d}{dx} \cos(x)^2=-2\cos(x)\sin(x)$$ $$\frac{d}{dx} \sin(x)^2+\frac{d}{dx} \cos(x)^2=0$$ $$\sin(x)^2+\cos(x)^2=c$$ ...
7
votes
2answers
102 views

Show $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$

If $\lim_{n \to \infty} a_{n} = a$ and $\lim_{n \to \infty} b_{n} = b$, how can we show that $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$? I say $\min\{a_{n},b_{n}\} $ has two cases: ...
0
votes
0answers
92 views

prove that a big-o estimate is correct for a pair of functions

Please could someone review my proof of the following big-O estimate thanks $(n^2+8)(n+1)$ f(n) is O(g(n)) if there are positive constants C and k such that: (1)f(n) $\le Cg(n)$ whenever n>k ...
3
votes
1answer
924 views

Every finite set contains its supremum: proof improvement.

Every finite subset of $\mathbb R$ contains its supremum (and its infimum) Proof Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb{R}$. Since it is non-empty and it is bounded ($\max A$ is ...
1
vote
2answers
557 views

Evaluation of Derivative Using $\epsilon−\delta$ Definition

Consider the function $f \colon\mathbb R \to\mathbb R$ defined by $f(x)= \begin{cases} x^2\sin(1/x); & \text{if }x\ne 0, \\ 0 & \text{if }x=0. \end{cases}$ Use $\varepsilon$-$\delta$ ...
1
vote
1answer
312 views

Proof correctness Eigenvalues and Isomorphisms

Prove that $\lambda$ is an eigenvalue of $T \iff$ the map represented by $T-\lambda 1$ is not an isomorphism. Proof: $\rightarrow$ Suppose $\lambda$ is an eigenvalue of $T$, then we have ...