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.

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0
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1answer
95 views

Contradiction proof for a limit law $f(x) \le g(x)$

Suppose that $f(x) \le g(x)$ for all $x$. Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$, provided these limits exist. I posted a similar question, but this is a different ...
2
votes
2answers
37 views

Proof by Induction - Wrong common factor

I'm trying to use mathematical induction to prove that $n^3+5n$ is divisible by $6$ for all $n \in \mathbb{Z}^+$. I can only seem to show that it is divisible by $3$, and not by $6$. This is what I ...
0
votes
0answers
26 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
5
votes
1answer
102 views

With $N$ a constant $>0$, show $\prod_{p<x}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{\substack{p<x \\ p \ \text{prime}}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this ...
0
votes
3answers
57 views

Proof by minimum counter example

I need to prove that $n^4-n^2$ is divisible by 12 by minimum counter example. I understand the process but I don't understand how we arrive at m>=7. I have seen different proofs but I still don't know ...
7
votes
3answers
109 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
0
votes
0answers
26 views

What theorems are used in this following proof of derivatives of log normalizer is moments of sufficient statistics?

The below is the derivation of the proof that shows derivative of log normalizer of exponential family is moments of sufficient statistics \begin{equation} ...
0
votes
2answers
51 views

GCD proof using fundamental theorem of arithmetic

prove: $\gcd(m,n)=1$ if and only if $\gcd(m^i,n^r)=1$ I believe you need to do something with fundamental theorem of arithmetic to prove one of the sides. Not quite sure though. Help is appreciated. ...
-2
votes
1answer
49 views

Combinatorial Argument Proof

Prove: $c(40,5) = c(17,5) + c(17,4) + c(23,1) +...+ c(23,5)$ where c is the binomial coefficient. Can I use a combinatorial argument to prove?
0
votes
2answers
51 views

Proof using Induction

Give the induction proof of: $$ k(k+5) = \frac{k}{5} $$ Is this proof even possible? Not sure how to do.
0
votes
1answer
25 views

Proof between max independent set cardinal and min vertex cover.

i'm tryign to solve this problem for my graph class, but I don't really know where to start. Be G a graph without isolated vertex,proof that it verifies that $\alpha \leq \beta$, where $\alpha$ is ...
2
votes
1answer
52 views

Finding a proof or a counter example in a programming puzzle

Some years ago I entered a programming contest and this was one of the problems: Binary Granny Summary: Given a positive integer N find 2 positive integers such that $$ x + y = N $$Let X and Y be the ...
1
vote
1answer
35 views

How to show there are infinite solution to a given Pell's equation?

I was asked to prove the Pell's equation $$x^2-7y^2=1$$ has infinitely many solution. Here is what I did By using Brahmagupta method we can generate infinitely many integer solutions. Is that ...
0
votes
2answers
40 views

Define f : Z/3Z → Z/3Z by f ([a]) = [2a + 1].

For this problem, I have to prove the function is well-defined, is surjective, and is injective. For seeing it is well defined, I have this: Assume [a1] = [a2] in the set of equivalence classes Z/3Z. ...
0
votes
1answer
26 views

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$.

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$. Prove that $f$ is well-defined, surjective and injective I don't really have a problem with figuring out if it's ...
1
vote
2answers
61 views

Finding a counterexample to a function proof

This is my proof: If f and g are surjective, then g ◦ f is surjective, with f: A $\to$ B and g: B $\to$ C. I have successfully proved this, but now I have to disprove the converse by finding a ...
1
vote
1answer
64 views

Problems Proving Injectivity and Surjectivity

I have these two functions, in which I have to prove or disprove they are injective and surjective: $f:[0,\infty) \to (0,\infty)$ by $f(x) = \frac{1}{x+1}$. $h:\mathrm R \to \mathrm R$ by $h(x,y) = ...
4
votes
2answers
62 views

Is $g : \mathbb R →\mathbb R$, $g(x) = |x|$ one-to-one and onto?

So, here is my function, in which I am to prove or disprove both if it is onto and one-to-one: Define $g : \mathbb R →\mathbb R$ by $g(x) = |x|$. For onto, can I say that it is not, because if we ...
0
votes
2answers
114 views

Justify each step in the proof sequence

$[A \rightarrow ( B \lor C) ] \land B' \land C' \rightarrow A'$ I know how to read the proof sequence, but I don't know what it means to "justify" each step? Does this mean to just state what each ...
3
votes
2answers
52 views

GCD Direct Proof

I want to try and prove this directly because I think it will be more straightforward then a indirect. Also, I believe this has something to do with relatively prime numbers. The help is appreciated! ...
1
vote
0answers
44 views

Working with the Mobius transformatios and linear algebra.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
0
votes
1answer
43 views

Let $\gamma$ be the Euler-Mascheroni constant. Can there be natural numbers $a,b,c$ such that $\log a - \log b - \log \log \log c =\gamma$?

Can there be integers satisfying $$\ \log a - \log b - \log \log \log c = \gamma \ \ \ ? $$
0
votes
1answer
29 views

If $x_k ≥ 0\;\forall \in \mathbb N$, and $y_k$ a bounded sequence, then the series $\sum_{k=1}^\infty x_ky_k$ converges

Hi I'm really struggling with this proof. For a start I'm struggling to believe it's true: For example, if we take $x_k = \dfrac{1}{k^2}$ and $y_k = -k^3$ (which is bounded above by any positive ...
0
votes
1answer
36 views

Counting sets and adding an element

Let $A$ be a set with $n$ elements, where $n \in \mathbb{\omega}$. Suppose $s \notin A$, prove that $A \cup \{s\}$ has $n+1$ elements. Here is what I have done so far: By induction, let $P(n):$ if ...
0
votes
1answer
90 views

Number theory practice exam questions

Looking for some help for these two practice problems for my exam. I'll explain to you what I have so far and my ideas. So for this problem, I solved part (a) using induction, it wasnt too tricky. ...
0
votes
0answers
47 views

Proving the limit comparison test

I have the next attempt: Because $0<L< \infty$, we can find two positive and finite numbers, $m$ and $M$, such that $m<L<M$. Now, because $L = lim_{n\to \infty} \frac{a_{n}}{b_{n}}$ we ...
1
vote
0answers
46 views

Is my proof right of this result?

Suppose that: $\sum_{n=1}^{\infty}a_{n}$ converges absolutely and $\{b_n\}$ is bounded.Prove that $\sum_{n=1}^{\infty}a_{n}b_n$ converges absolutely. My attempt: Let $M$ be the upper bound of ...
2
votes
2answers
64 views

Verifying the convergence of a series.

I need to prove that the series $$\sum_{n=0}^{\infty}3^{-n}$$ converges and to find the limit. My attempt: We can express our series as: ...
-1
votes
1answer
32 views

Prove $|A| \le|C|$ for injection and surjective functions

$A$, $B$ and $C$ are finite sets with $F: A \to B$ a surjection and $G: B \to C$ an injection. Prove $|A| \le |C|$ I could prove it using examples, but not sure how to generally.
2
votes
4answers
110 views

Proof by Geometric Intuition may Fail!!

In a book on differential forms, I read, "After all, there were times when people took geometric intuition as proof, and later found that their intuition was wrong". I would like to see an ...
0
votes
1answer
63 views

Prove $\sin x=3x-2$ has only one real solution

Obviously you can draw a graph, but how would you prove this with calculus?
2
votes
3answers
62 views

Combinatorial Argument with Natural Numbers

Give a combinatorial argument to show that all natural numbers c(n,k) = c(n,m) where c stands for combination.
1
vote
0answers
39 views

Prove that a set with n elements in union with an element not in the set has n+1 elements

Suppose $A$ has $n$ elements and suppose $a \notin A$, prove that $A \cup \{a\}$ has $n+1$ elements. I am pretty sure that I am supposed to use something with one to one and onto functions. It all ...
0
votes
1answer
34 views

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges. Also assume that the sequence is positive. lim sup$_{n} n^{2}a_{n} = 0$ means that for every $\epsilon$, ...
1
vote
1answer
26 views

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges The solution proof goes like: lim inf$_{n} na_{n} > 1 \Rightarrow$ there exists an $N \in \mathbb{N}$ such ...
1
vote
3answers
39 views

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$ Should I prove this by induction? If so, how should I go about it?
1
vote
1answer
112 views

The fix points of the Möbius transformations are the eigenspace of a certain matrix.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
1
vote
3answers
42 views

Proving quadratic inequalities?

I am trying to prove that $$e^{k+1} ≥ 3 + 3k + k^2$$ with, $$k>2$$ WhatI have done so far: What we are trying to prove is that $$e^n≥1+n+n^2$$ is a true statement. Since $n=3$ holds, this is ...
0
votes
1answer
24 views

Prove $|A| \leq |B|$ for $1-1$ function.

Prove $|A|\leq |B|$ if function $F:A\rightarrow B$ is a $1-1$ function. I wanted to know how to prove this out of curiosity. The help is appreciated.
-1
votes
3answers
76 views

Find $\bigcap_{n=1}^\infty(0,1/n)=\emptyset$ [closed]

I`ve tried this and is it true or completely not? Then how can I fix it? Proof: too wrong so I get it off
0
votes
1answer
47 views

Set difference bijection proof (stuck on injective)

So here is the question I'm working on So obviously I need to prove two things, that its both injective and surjective, however I'm trying to show it is injective and am currently stuck, here is what ...
0
votes
1answer
34 views

How to prove that a set is not totally ordered?

I know that a set to be totally ordered and for example $A,B \in P(X)$ must either be $A \le B$ or $B \le A$. And also $\le$ is equivalent to $\subset$ for sets. But I am not sure how I would prove ...
0
votes
0answers
59 views

Proving symmetry of metric (single linkage between clusters using arbitrary dissimilarity measure)

I am told to assume that our dissimilarity measure $d$ satisfies the properties required of it, what seems to be the definition of a metric: $d(x,y) \geq0 $ and $d(x,y)=0 \Longleftrightarrow x=y$ ...
0
votes
0answers
55 views

Show f is integrable and integral is C(b-a)

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$ Consider for real $a<b$ and real $A,B,C$, the function ...
2
votes
1answer
42 views

Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function.

Let $H$ and $K$ be normal subgroups of a group $G$, with $H \subseteq K$. Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function. [That is, if $Ha=Hb$, ...
1
vote
4answers
111 views

Proof that $f(x)=\frac{(\sin x)^3}{x}$ gets maximum in $(0,\infty)$.

I have this problem, and I got stuck in my proof Prove $f(x)=\dfrac{(\sin x)^3}{x}$ gets maximum in $(0,\infty)$. My Proof $$(1)\lim_{x \to 0+} \frac{(\sin x)^3}{x}= 0$$ $$(2)\lim_{x \to \infty} ...
0
votes
1answer
29 views

How to proove that smallest upper bound exists und it is cleary determined?

Let X be a set. Then a relation '$\le$' on $\mathcal P(X)$ is defined by: $A \le B :\Leftrightarrow A \subset B$ . Let $\mathcal A \subset \mathcal P(X)$. One set $B \in \mathcal P(X)$ for which ...
1
vote
1answer
44 views

Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
0
votes
1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
0
votes
1answer
19 views

Understanding how to prove a bijection into three sets

I understand how to prove if there is a bijection from A onto B. However, say that there is a bijection from A onto B and a bijection from B onto C. How would I prove that that there is a bijection ...