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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0
votes
2answers
104 views

Suppose f : A ---> B and g : B ---> A are functions for which g o f = 1A…

If I were to suppose that $f : A \to B$ and $g : B \to A$ are functions for which $g \circ f = 1_A$, is $f$ always surjective and is $g$ always injective? How would I either prove this or counter it? ...
0
votes
1answer
26 views

Having trouble with showing this CANNOT be a theorem in incidence geometry.

Consider the following statement: If l and m are any two distinct lines, then there exists a point P that does not lie on either l or m. (a) Show that this cannot be a theorem in incidence geometry. ...
6
votes
5answers
256 views

Verify the following combinatorial identity: $\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$ [duplicate]

$$\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$$ Nice, so I've proven some combinatorial identities before via induction, other more simple ones by committee selection models.... But ...
2
votes
1answer
122 views

About using Archimedean property to prove the existence of least upper bound

I am studying the proof of existence of least upper bound, but I can not understand how the autor applies the Archimedean property. Archimedean property. Let $x$ and $\epsilon$ be any positive ...
4
votes
1answer
94 views

Prove that the kernel is of dimension 2

"Experimentally", I found that the kernel (null space) of the following matrix is of dimension 2. I'd like to prove it, but haven't managed yet: \begin{equation} \text{for almost all } t>0,\quad ...
0
votes
2answers
33 views

Prove If LCM(a,b) = c and a|k and b|k then c|k.

Prove If LCM(a,b) = c and a|k and b|k then c|k. I know that c divides a and b if c = Least Common Multiple of a and b. I also know that c divides all multiples of a and b. I just am not sure how ...
0
votes
2answers
44 views

Prove this sum of binomial terms using induction.

Here's the problem stumping me today: Let $n \in \mathbb{N}$ and $r \in \mathbb{N}$ such that $r \leq n$, and prove using induction that $\binom{n+1}{r+1} = \sum\limits_{i=r}^n \binom{i}{r}$. I've ...
0
votes
2answers
40 views

Suppose that the $\lim a_{n} = 1$ and $x < y$

Suppose that $\lim a_{n} = 1$ and $x<y$. Is it possible to show using the Algebriac limit theorem that if the $\lim xa_{n} < y$. Then $0<x<y$? .
1
vote
0answers
90 views

no. of regions a plane is divided into by $n$ lines in general position

My notes state the Counting process for knowing no. of regions a plane is divided into by $n$ lines in general position := Let $h_1(n)=$ No. of parts a line is divided by $n$ distinct ...
4
votes
1answer
69 views

Prove [(xy=0)∧(x,y∈ℤ)]→[(x=0)∨(y=0)]

I'm working through a higher algebra textbook. It has some exercises related to the positive integers and I'm stuck on this proof. Here's what I have so far: Attempted proof Assume the contrary, ...
2
votes
1answer
20 views

$[x]_0,[x]_1\ldots [x]_n$ is a basis for vector space $V$.

here is a lemma which requires the use of falling factorials which are written as $[x]_n=x(x-1)\ldots(x-(n-1))$ : Lemma:Let $V$ be a vector space of polynomials over $\mathbb C$ , then ...
0
votes
1answer
51 views

show that if det()=0 with A, and B, then AB = BA.

show that if |b c| |a b| = 0 with A = [a a] [b b] , and B = [b b] [c c], then AB = BA. I'm not exactly sure how to go about proving this. I tried computing both AB and BA, then factoring ...
0
votes
1answer
58 views

Proofs in Stochastic Processes

Let $$X_{n}$$ be an irreducible Markov chain on the state space {1,...,N}. Show that there exists $$C < \infty$$ and $$\rho < 1$$ such that for any states i,j, $$\mathbb{P} [ X_{m}\neq j , m=0 ...
0
votes
1answer
52 views

The Completeness Axiom: Proof

So far I only did a because I wasn't sure about my proof and I wanted to make sure that made sense first: Let a/b be a fraction in lowest terms with $$0<a/b<1$$ a. prove that there exists ...
3
votes
3answers
52 views

Prove $(\log{n})^2\leq 2^n$ by induction

I've trying to solve this for quite a while now, but not being able to finish the proof. Prove using induction that $(\log{n})^2\leq 2^n$
3
votes
6answers
109 views

Prove that if $n^2$ is even then $n$ is even

Assume that $n^2$ is even Therefore $n^2 = 2k$ for some integer $k$. How do I finish this proof?
2
votes
1answer
30 views

How do I *formally* get informations from a presentation for a group?

Just for clarification, here is the definition for a presentation for a group: Let $G$ be a group and $S$ be a set. Let $F(S)$ be the free group on $S$ and $R\subset F(S)$ and $\overline{R}$ ...
1
vote
1answer
44 views

Combinatorial or algebraic proof

I am having trouble proving this identity using combinatoric or algebraic proof. As someone pointed me out it is somehow related to pascals triangle recurrence. $$\sum_{i=0}^k \binom{n+i}{i} = ...
3
votes
0answers
63 views

Random Wolfram|Alpha identity $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$

I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series $$\sum_{k = ...
0
votes
1answer
49 views

Proof using Mean Value Theorem: $f(a)=g(b), f(b)=g(a)$. Prove $f'(c)g'(c)\leq 0$

Can someone help me with this problem please? Assume that the functions $f$ and $g$ are differentiable on $[a,b], f(a)=g(b)$ and $f(b)=g(a)$. Prove that there exists $c \in (a,b)$ so that $f'(c)g'(c) ...
1
vote
1answer
67 views

If the real part of $f$ is bounded above by $A$ in the unit disk, then $|f(z)| \leq {2A|z|}/({1-|z|})$

Let $f$ be a holomorphic function on unit disk $\mathbb{D}$ such that $f(0)=0$ and there exists constant $A>0$ such that $\operatorname{Re}(f(z)) \leq A$, for all $z \in \mathbb{D}.$ Could anyone ...
8
votes
1answer
651 views

A long nasty limit problem

Does the following limit admit a closed-form? $$\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$$ My professor gives this ...
0
votes
1answer
32 views

Can we find a real $u$ such that $f(u)=w$ ($w$ is fixed) and $f'(u)≠0$?

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real ...
1
vote
1answer
64 views

Power Series of Recurrence

Let n be a Natural number. Define $\ S_n $ to be the set of compositions of $\ n $ where no part is equal to 2, and let $\ a_n = |S_n| $. It is trivial that: $$ a_n = [x^n] \frac{1-x}{1-2x+x^2-x^3} ...
1
vote
1answer
113 views

Delta Epsilon Quotient Proof

I'm trying to justify $\lim_{x\rightarrow x_0} \frac{x^3-8}{x-2} = 12$, where $x_0=2$ using $\delta-\epsilon$ proofs. I've never done one of these proofs where $\lim_{x\rightarrow x_0} \frac{L_1}{L_2} ...
2
votes
3answers
66 views

Linear Maps: Prove if $T^2 =0$, then $I-T$ is bijective

Let $V$ be a vector space, $T$ is in $L(V)$, Prove: If $T^2 = 0$, then $I - T$ is bijective. the book also gave a hint: in polynomial algebra, $(1-t)(1+t)=(1-t^2)$ I'm not quite sure where to start. ...
0
votes
0answers
47 views

Is this metric complete?

If $\mathbb{R}$ uses $d(x,y) = |x^3 - y^3|$, is this complete? I thought that if $(x_n)$ be Cauchy, then $|x_n^3 - x_m^3| < \epsilon$ for $n, m > N_1 \in \Bbb N.$ I need to show ...
0
votes
1answer
129 views

Prove using a proof by contradiction: There is no smallest positive real number

Prove using a proof by contradiction: There is no smallest positive real number Let us assume the contraction: There is a smallest positive real number. How do I continue?
4
votes
4answers
378 views

Mathematical Induction Question, Proof Help [duplicate]

Prove using Mathematical Induction that for all natural numbers ($n>0$): $$ \frac 1 {\sqrt{1}} + \frac 1 {\sqrt{2}} + \cdots + \frac 1 {\sqrt{n}} \ge \sqrt{n}. $$ ...
0
votes
3answers
31 views

Let n be an integer. Prove that if $2|(n^2-1)$ then $4|(n^2-1)$.

I know that $n^2=2k$ for some integer $k$. Please help me continue. i've got a midterm tomorrow that I'm really stressed about. Thanks
0
votes
5answers
60 views

Let $n $ be an integer. Prove that if $ 2|n$ and $3|n$, then $6|n$ [closed]

Let $n $ be an integer. Prove that if $ 2|n$ and $3|n$, then $6|n$ How do I prove this? I know that: $n =2k$ for some integer $k$, $n = 3q$ for some integer $q$
0
votes
0answers
75 views

Prove an implication about quadratic form definiteness

Sylvester's criterion states that a quadratic form $q$ over an $n$-dimensional real linear space $V$ is positive definite $\iff$ all main minors $\Delta_1, \Delta_2, ..., \Delta_n > 0$. In ...
2
votes
1answer
47 views

Convergence of $\sum_{n=0}^\infty (-1)^n (e-(1+\frac{1}{n})^n)$

Does $\sum_{n=0}^\infty (-1)^n (e-(1+\frac{1}{n})^n)$ converge absolutely, conditionally, or diverge? Attempt: Yes, by the ratio test we have $$ \lim_{n \to \infty} \left| \frac{(-1)^{n+1} ...
2
votes
1answer
146 views

Proof with combinatorial argument

Show with combinatorial argument that this is equal : $$\dbinom{n}{k+1} = \dbinom{n-1}{k}+ \dbinom{n-2}{k} +...+ \dbinom{k}{k}$$ I have no idea how to do that so it would be really helpful ...
0
votes
2answers
19 views

Proof on primitive elements

'Suppose x is an element of order $\phi$(n) in $\mathbb{Z}$/n$\mathbb{Z}$. Then every invertible element of $\mathbb{Z}$/n$\mathbb{Z}$ is a power of x.' The lecture taught me that when this ...
3
votes
3answers
271 views

If $f$ is continuous on $\mathbb R$ then $\exists c\in\mathbb R: f(x)=c$ has only one solution

I have to prove that there is no continuous function $f: \mathbb{R} \to \mathbb{R}$ such that, for each $c \in \mathbb{R}$ the equation $f(x)=c$ has exactly two solutions. My attempt: We have that ...
2
votes
4answers
141 views

Prove that a continuous function defined on an interval $[a,b]$ has a fixed point.

I have to prove that : Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$. But I don't know how to attack it since I can't apply anything of ...
1
vote
1answer
37 views

Prove the uniformly continuity of a function with a certain property

I need to prove this: Suppose that $f: \mathbb{R} \to \mathbb{R}$ is continuous and has the property that for each $\epsilon >0$ there is $M>0$ such that if $|x| \ge M$, then $|f(x)|< ...
0
votes
3answers
76 views

Prove that the set $A:=\{ x : 0 \le f(x) \le 1 \}$ is compact.

I have to prove the following: Suppose that $f : D \to R$ is continuous with $D$ compact. Prove that $\{ x : 0 \le f(x) \le 1 \}$ is compact. My attempt: We define $$A:=\{ x : 0 \le f(x) \le 1 ...
0
votes
0answers
74 views

Proof by Induction for Splay Tree?

I'm preparing for an exam about Trees. One of the questions that appear in Mark Allen Weiss' "Data Structures and Algorithms Analysis in C++" is: Prove by induction that if all nodes in a splay tree ...
1
vote
2answers
177 views

Prove that there exist two integers such that i - j is divisible by n.

Here's the full question: Prove that, for any $n + 1$ integers, $\{x_0, x_1, x_2, . . . , x_n\}$, there exist two integers $x_i$ and $x_j$ with $i \neq j$ such that $x_i − x_j$ is divisible by $n$. ...
2
votes
0answers
56 views

Prove circle packing solution is optimal

Background: This is a follow on from this question of how to maximise the area of two non overlapping circles of arbitrary radii packed into a rectangle of arbitrary width and height. I proposed a ...
1
vote
3answers
386 views

Show that there are no positive integer solutions to x^2 + x + 1 = y^2.

I'm trying to prove that $x^2 + x + 1 = y^2$ has no integer solution, but I'm having a lot of trouble. So far I've tried proof by contradiction, but all of that seems to rely on me being able to ...
3
votes
2answers
126 views

The Conjecture That There Is Always a Prime Between $n$ and $n+C\log^2n$

Let $n$ be 113. Use $n+C\log^2n$ to find the next consecutive prime or at least approximately how far away it is. Will you show me how to work this out step by step to show me how to use this formula? ...
0
votes
2answers
29 views

Homomorphisms between fields are injective.

How would I prove this? I know that I must show f(a)=f(b) => a = b I also know I must use the definition of homomorphism, ie: $f(a+b)=f(a)+f(b)$ $f(ab)=f(a)f(b)$ $f(1)=1$ I am assuming that a ...
1
vote
1answer
38 views

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$.

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$. My Attempt: $(\rightarrow):$ $$\alpha i=ai-b-ck+dk \Rightarrow -b+ai+dj-ck$$ ...
0
votes
1answer
31 views

Differentiation proof

Find the co-ordinates of the point on a curve $y=x^2+3x-1$ at which it is parallel to the line $ y=5x-1?$ unsure how to solve this
0
votes
1answer
54 views

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$.

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$. First, $\mathbb Q(z)\subseteq \mathbb Q(z^3,z^5)$ would be trivial, right? Then we ...
2
votes
2answers
39 views

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$.

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$. I understand that $\mathbb Q(\sqrt{a})$ is the smallest ...
4
votes
2answers
110 views

Let $x, y \in \Bbb Z$. If $x + y \geq 135$, then $x > 67$ or $y > 67$.

Let $x, y \in \Bbb Z$. If $x + y \geq 135$, then $x > 67$ or $y > 67$. How do I prove this statement? I'm new to proofs, and I find this to be too obvious to prove. Please help me