For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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2
votes
1answer
33 views

Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
0
votes
2answers
27 views

Proof of the inequality $F_i<(5/3)^i$ for the Fibonacci numbers

The example states: As an example, we prove that the Fibonacci numbers, F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., Fi = Fi - 1 + Fi - 2, satisfy Fi < (5/3)i, for all i >= 1. To do this, we ...
0
votes
2answers
30 views

Proving the arithmetic mean equals the geometric mean when $a=b$.

Arithmetic mean $a,b \in \mathbb R$ is $A(a,b)=\frac{a+b}{2}$ Geomtric mean $a,b \in\left[0,\infty\right]$ is $G(a,b)=\sqrt{ab}$ I'm trying to prove that $G(a,b)=A(a,b)$ if and only if $a=b$. ...
1
vote
1answer
39 views

'Identity theorem' for Meromorphic functions

If $f_1,f_2$ are meromorphic functions in $D$ and there exists a sequence of pairwise distinct points $z_n \in D$ such that $z_n \to z_o \in D$ and $f_1(z_n)=f_2(z_n),$ then $f_{1} \equiv f_2$ on $D.$ ...
0
votes
2answers
28 views

Convergence of integral, that is absolutely convergent, proof

Can you think of any proof on convergence of improper integral, that is absolutely convergent? It is so obvious, that I really don't know where to start. Triangle inequality gives us ...
2
votes
1answer
24 views

Radius of convergence of sum of complex power series

Could anyone advise me on how to find radius of convergence of $\sum^{\infty}_{n=1} [\frac{1}{n^2}+(-2)^n]z^n \ ?$ Thank you. My attempt: radius of convergence of $\sum^{\infty}_{n=1} ...
0
votes
1answer
27 views

Proving $\mathrm{Hom}(V \rightarrow W)$ is a vector space

It can easily be proven that $\newcommand{\Hom}{\mathrm{Hom}}\Hom(V \rightarrow W)$ is a sub-space. 1. we know that for any $T:V\rightarrow W$, T(0)=0, therefore $0\in \Hom(V \rightarrow W)$ 2. ...
0
votes
0answers
11 views

Inclusion of commutators on classical pseudodifferential operators

We denote by $Cl^\mu$ the class of classical pseudo-differential operators of order $\mu$. Consider the notation $$[Cl^{a},Cl^{b}]\hookrightarrow [Cl^{a'},Cl^{b'}]$$ which means that a commutator on ...
4
votes
0answers
33 views

Finding an analytic function such that real part is the given function.

I am reading the book Complex Analysis by Lars V Ahlfors. In the book he uses a nice method without involving integration to evaluate $f$ given that the real part of the function is $U$. The method ...
0
votes
0answers
17 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
0
votes
0answers
7 views

Convex combination of polynomials with roots on the unit circle and companion matrix

Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ ...
1
vote
2answers
49 views

Sequences and series

If $p, q, r$ are in G.P. and the equations: $$px^2 + 2qx + r = 0$$ $$dx^2 + 2ex + f = 0$$ Have a common root, then show that $$\frac{d}{p}, \frac{e}{q}, \frac{f}r$$ are in A.P. Well I tried taking ...
1
vote
2answers
62 views

Exercise about truth functions in J.R.Shoenfield's “mathematical logic”

The first exercise in Joseph R. Shoenfield's "mathematical logic" is: An n-ary truth function $H$ is definable in terms of the truth functions $H_1,\dots,H_k$ if $H$ has a definition ...
2
votes
4answers
70 views

Prove that $133$ divides $11^{n+1} + 12^{2n−1}$ for all $n > 1$? [duplicate]

I have tried this question so hard but still stuck here. It seems like easily provable if all $n$ are all positive numbers but in this question, the $n$ is bigger than $1$. original question : prove ...
-2
votes
1answer
60 views

why is (a proof of 1+1=2) necessary? [on hold]

http://mathforum.org/library/drmath/view/51551.html Now, empirically speaking, I have this one on the left, and that one on the right, and together they make these two right there. why is it ...
1
vote
0answers
27 views

Proof Verification: For $f: A \to B$ and $T \subset B$ show that $f^{-1}(T') = (f^{-1}(T))'$.

I want to know if my proof is correct and if the strategy I use should be used for all questions of this form. Compliments are taken with respect to the set $B$. My method of proof would be to first ...
0
votes
2answers
94 views

Does there exists an entire function with the following property: $f\left(\frac{1}{n}\right)= \frac{n^4}{1+n^4}, n =1,2,…$

Could anyone advise me on how to use the Identity theorem to determine whether there exists an entire function with the following property: $f\left(\dfrac{1}{n}\right)= \dfrac{n^4}{1+n^4}, n =1,2,...$ ...
2
votes
1answer
69 views

Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite

For each real number $x$, let $[x]$ be the largest integer less than or equal to $x$. For example, $$[5] = 5$$ $$[7.9] = 7,$$ and $$[−2.4] = −3.$$ An arithmetic progression of length $k$ is a ...
1
vote
1answer
28 views

If $(y_n(x))_{n \in \mathbb{N}}$ is uniformly convergent, so is $(f(x,y_n(x)))_{n \in \mathbb{N}} \ ?$

Let $f$ be a continuous function defined on $[a,b] \times [c,d]. $ Consider $(y_n(x))_{n \in \mathbb{N}}$ such that it is uniformly convergent on $[e,f] \subseteq [a,b].$ Could anyone advise me on ...
1
vote
2answers
45 views

Proving that $\sqrt{pq} \ne (p + q)/2$ implies $p \ne q$ using the contrapositive

Prove by the contrapositive method, that if $p$ and $q$ are positive real numbers with the property that $\sqrt{pq}$ is not equal to $(p+q)/2$, then $p$ is not equal to $q$. The proof is easy enough ...
4
votes
1answer
56 views

To show that a concretely defined group is isomorphic to an explicitly presented group, what strategies are available?

I have a homework problem of the following form. We're given presentation of a group $\langle x,y \mid R\rangle$ explicitly, and two matrices $X,Y \in \mathrm{GL}(\mathbb{C},2).$ We know $X$ and $Y$ ...
2
votes
1answer
60 views

Prove $ x^n-1=(x-1)(x^{n-1}+x^{n-2}+…+x+1)$

So what I am trying to prove is for any real number x and natural number n, prove $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$$ I think that to prove this I should use induction, however I am a bit stuck ...
1
vote
1answer
21 views

Proof using vectors - trigonometric formulas

Question: If two vectors a and b make angle $\alpha$ and $\beta$ with the x-axis, prove, using vectors, that: $$\cos(\beta - \alpha) = (\cos \alpha) (\cos\beta) + (\sin\alpha) (\sin\beta)$$ I tried ...
3
votes
4answers
53 views

Proof by contradiction that $(n+1)^3 \not= n^3 +(n-1)^3$ for $3$ consecutive positive integers

Prove by contradiction that if n-1, n, n+1 are consecutive positive integers, then the cube of the largest cannot be equal to the sum of the cubes of the other two. Assume that: $$ (n+1)^3 = ...
2
votes
1answer
28 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
50
votes
8answers
2k views

Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, and then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, ...
1
vote
2answers
42 views

If a continuous function is positive at a point, it is also positive in some neighborhood of the point [closed]

Suppose that $f:\mathbb{R}^k\to\mathbb{R}^1$ is a continuous function and that $f(x^*)>0$. Show that there is a ball $B=B_\delta(x^*)$ such that $f(x)>0$ for all $x\in B$.
0
votes
1answer
26 views

Prove: If this system is solvable, then this dual system is not.

I'm trying to get a handle on algebraic dual spaces, and it's hurting my head. To be proven: Let $A$ be a $m \times n$-matrix and $b$ be a $1 \times n$-matrix. Show that the system $$\begin{cases} ...
0
votes
1answer
105 views
+50

Understanding last step of a proof that “two trajectories cannot cross at a finite value of t” (Phase trajectories/nodes)

Note: This proof prefaced critical points at the origin for coupled first order ODEs. It was done before showing the asymptotically stable and unstable critical points: Improper, Proper, Spiral, ...
1
vote
2answers
117 views

Is there a more direct way of proving that this ring is an integral domain?

In self studying abstract algebra and I've come upon the following problem which I could not solve directly. For any $d\in \mathbb{Z}$ we are asked to show that $\mathbb{Z}[\sqrt d]=\{a+b\sqrt{d} ...
1
vote
1answer
42 views

Divisibility problem ($p \leq \sqrt{n}$)

If $n \geq 2$ and $n$ is composite, then there exists a prime $p$ such that that $p \mid n$ and $p \leq \sqrt{n}$ As $n$ is composite, it follows that $n = ab$ for some $a, b \in \Bbb N$, where ...
3
votes
2answers
39 views

Prove that there is no bipartite graph on $14$ vertices with this degree sequence.

Prove that there is no bipartite graph on $14$ vertices with degree sequence: $$6, 6, 6, 6, 6, 6, 6, 6, 5, 3, 3, 3, 3, 3.$$ I assume a vertex with degree $5$ breaks this graph from being ...
0
votes
0answers
36 views

Proof Strategy: Induction Summation of Series

Let $P(n)$ be the following statement: $$\sum\limits^{n}_{i=0}r^i = \dfrac{1-r^{1+n}}{1-r}\text{ for all }n \in \mathbb{N}\text{.}$$ I am stuck at the base case: $$P(1):1 + r = ...
0
votes
1answer
37 views

How to prove the uniqueness of linear functional

$\textbf{Theorem}$ If $V$ is a $n$-dimensional vector space, if $\{x_1,.,.,., x_n\}$ is a basis in $V$ and if $\{\alpha_1,\cdots \alpha_n\}$ is any set of $n$ scalars, then there is one and only one ...
1
vote
4answers
57 views

How to prove correctness of this algorithm

Situation There is a long patch of grass with seeds planted along it. Each seed needs to be within 2 metres of a sprinkler in order to be watered daily. Describe an algorithm that will result in ...
0
votes
1answer
25 views

Prove that for any $\{x_{m_k} \}\in I_n$, where $I_n$ are dyadic intervals, $\lim_{n \to \infty} x_{m_k} =c$

Proving for any $\{x_{m_k} \}\in I_n$ that:$$\lim_{n \to \infty}\{x_{m_k} \}=c$$ I have been trying to solve this problem, but i dont know how to write it properly, so i need your help whit ...
3
votes
2answers
60 views

How do you formulate a vague notion into a mathematical expression?

I am a software engineer wanting to learn math. I also do a little bit of drawing. What I am wondering is, how do you formulate a vague notion of something you're trying to model into a mathematical ...
2
votes
0answers
34 views

Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
1
vote
2answers
40 views

Doubt in proof of Dual of the direct sum

If $M$ and $N$ are subspaces of $V$, and if $V = M \oplus N$, then $$V' = M^\perp \oplus N^\perp$$ where $W^\perp$ is the annihilator of $W$. I didn't understand how to prove both of the ...
1
vote
1answer
37 views

If $n$ is any positive integer whose last digit is $5$, then $5$ divides $n$

Prove that if n is any positive integer whose last digit is a 5, then 5|n Therefore, n is going to be 5, 15, 25, 35 etc ... b∣a states that 'b divides a' and we know that 5∣5, 5∣15, 5∣25, 5∣35 ...
1
vote
0answers
15 views

Gaussian integral of a function with nonzero mean

From the wikipedia article, for a Gaussian integral of an analytic function we have that I'm trying to obtain a similar formula when there is a linear term in the Gaussian (ie the Gaussian has a ...
0
votes
3answers
87 views

prove transitivity property congruence mod m

Prove transitivity property of congruence mod m. Show that if $x\equiv y \pmod m$ and $y \equiv z\pmod m$ then $x\equiv z\pmod m$ . I didn't really get the tutors explanation of this, I get what ...
0
votes
5answers
36 views

Injective function proof involving floor function

Let $f : \Bbb{Z} \to \Bbb{Z}$ and $g : \Bbb{Z} \to \Bbb{Z}$ be functions defined by $f(x)=3x+1$ and $g(x)=\lfloor\frac{x}{2}\rfloor$. Is $g(f(x))$ one-to-one? So, $g(f(x)) = ...
2
votes
2answers
65 views

prove that any integer greater than or equal to 8 can be represented as the sum of nonnegative integer multiples of 3 and 5

This problem asks to use Well Ordering Principle to prove any integer greater than or equal to 8 can be represented as the sum of nonnegative integer multiples of 3 and 5. Here's where I'm at: For ...
0
votes
0answers
31 views

Problem: use the well ordering principle to show that all positive rational numbers can be written in lowest terms

This problem involves pointing out the unjustified inference/logic error in the following proof that all positive rational numbers can be written in "lowest terms" that is as a ratio of positive ...
0
votes
2answers
42 views

Help With a proof (Irrational Number)

Prove the following statement by proving its contrapositive: if $r$ is irrational, then $r^\frac{1}{5}$ is irrational. Its contrapositive will be: If $r^\frac{1}{5}$ is not irrational, then $r$ is ...
0
votes
1answer
23 views

How to prove the Hubble law is the unique expansion law compatible with homogeneity and isotropy?

In the book physical foundations of cosmology, it saids that Hubble law is unique and a problem seems to be a hint of proving that. In order for a general expansion law,v=f(r,t), to be the same ...
1
vote
0answers
32 views

From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
3
votes
3answers
47 views

Show surjectivity of a linear map

It pains me to say that this bewilders me, but here's the problem. All I want to do is show that: Given $T$ a linear operator on some finite-dimensional space $V$, with the property that $Im(T) = ...
1
vote
7answers
111 views

Error in proving of the formula the sum of squares

Given formula $$ \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$ And I tried to prove it in that way: $$ \sum_{k=1}^n (k^2)'=2\sum_{k=1}^n k=2(\frac{n(n+1)}{2})=n^2+n $$ $$ \int (n^2+n)\ \text d ...