Tagged Questions

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
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0answers
7 views

Continuity argument to show that the derivative exists everywhere.

I have shown that, for $f(X) = \det(X)$, $$\mathrm d f_A(H) = \mathrm{tr} (\mathrm{adj}(A) H)$$ But I have only show this for invertible $A$. I wish to use a continuity argument to show that this ...
1
vote
5answers
50 views

If $f'(0)$ exists then show that $f$ is differentiable everywhere.

Suppose $f'(0)$ exists and that $f(x+y)=f(x)f(y)$ for all $x\in\mathbb{R}$. Show that $f$ is differentiable for all $x\in\mathbb{R}$. I'm not exactly sure how to go about this. I'm not looking for a ...
2
votes
2answers
36 views

Proof by smallest counterexample

Actually what i know is that i must assume that this statement is false and then try to come up with non sense statement.
0
votes
1answer
15 views

Generalizing Cauchy's Criterion to $\mathbb{R}^p$

I'm trying to generalize the following statement about Cauchy's Criterion: A sequence in $\mathbb{R}$ satisfies CC $<=>$ it converges And I want to generalize it to CC of a sequence in ...
-1
votes
0answers
18 views

If $R$ is a ring, $S$ is a subring, and $I$ is an ideal of $R$, then $I\cup S$ is an ideal of $S$

Proof or Counterexample: If $R$ is a ring, $S$ is a subring, and $I$ is an ideal of $R$, then $I\cup S$ is an ideal of $S$. Just before this I encountered a similar problem with $I\cap S$ and was ...
1
vote
2answers
20 views

how to know when a particular proof is appropriate for the given problem?

The main trouble I am currently having in math is knowing when the use cases are appropriate in a proof. I see many videos where they seem to choose a strategy like proof by contrapositive or proof by ...
1
vote
1answer
30 views

Harnack's inequality

Let $u$ be harmonic on $\{|z|<1+\epsilon\}$ for some $\epsilon>0$ and $u \geq 0$ on $\{|z|=1\}.$ Could anyone advise me how to show $\dfrac{1-|z|}{1+|z|}u(0) \leq u(z) \leq ...
4
votes
4answers
96 views

Prove that $\sqrt{2} + \sqrt{3}$ is irrational. [duplicate]

Assume that $\sqrt{2}, \sqrt{3}, \sqrt{6}$ are all irrational. Prove that $\sqrt{2} + \sqrt{3}$ is irrational. So I know how to prove this using contradiction, and assuming that it is rational. But, I ...
-2
votes
0answers
12 views

Derive L = M / (m + M) from given properties.

mv = (M+m)V KE(1) = (1/2)m(v^2) KE(2) = (1/2)(m+M)(V^2) L = [KE(1) - KE(2)]/KE1 Prove that L = M/(M+m) Thank you!
0
votes
0answers
11 views

help with proof involving matrix derivations

So, Ive been trying to learn the research in a particular article, which can be read http://www.sciencedirect.com/science/article/pii/0024379580902219# Specifically lemma 2. So far, I have understood ...
1
vote
1answer
18 views

How to prove 120 degree rotations of a hexagon form a subgroup

Let H={$\rho_{0}, \rho_{2}, \rho_{4}$}, a subgroup of D6, the group of symmetries. Where $\rho_{0}$=identity permutation, $\rho_{2}$=(1,3,5)(2,4,6) and $\rho_{4}$=(1,5,3)(2,6,4) Identity is easy to ...
0
votes
0answers
30 views

How to prove isomorphism between these two graphs

I thought that the best way to approach this problem was to use a direct proof and say that since the graphs have the same number of vertices G1: {v1, v2, ..., vi, ..., vk} and G2 : {b1, b2, ..., ...
3
votes
2answers
40 views

There is no holomorphic function in $\Omega=\{0<r<\lvert z\rvert <R\}$ with real part $u(x,y)=\frac{1}{2}\log(x^2+y^2)$

Consider $u(x,y)=\dfrac{\text{log}(x^2+y^2)}{2}$ on $\Omega=\{0<r<|z|<R\}.$ Show there is no holomorphic function on $\Omega$ whose real part is $u.$ My attempt: I understand that $u$ ...
0
votes
1answer
40 views

Prove the volume of a ball with radius approaching 0

Let f be continuous and let Br be the ball of radius r > 0 centered at $(x_0, y_0, z_0)$. Let V (Br) be the volume of Br. Prove that $$\lim_ {r\to0} \frac{1}{V(Br)}\ \iiint_{Br} \ f(x,y,z) dV = f(x_0, ...
0
votes
1answer
19 views

Proving a point exists on a twice differentiable function.

Problem: Suppose that $f$ is twice differentiable on $(a,b)$ and that there are points $x_1\lt x_2\lt x_3$ in $(a,b)$ such that $f(x_1)\gt f(x_2)$ and $f(x_3)\gt f(x_2)$. Prove that there is a point ...
3
votes
2answers
21 views

Division with remainder

I have proved the division with remainder theorem: If a $\in \mathbb{Z}$ and $d \in \mathbb{N}$ then there exists unique numbers $q,r \in \mathbb{Z}$ such that $a=dq+r$ where $0\le r<d$. I proved ...
2
votes
0answers
19 views

Find the initial movement of a particle

A particle with mass $m$ is moving along a curve and the force exerted on it always points towards the origin, and it´s magnitude is proportional to the distance between the particle and the origin, ...
2
votes
2answers
26 views

Proving that a graph is connected?

I'm trying to prove that this graph is connected given the provided information. Let $G$ be a simple undirected graph with $n \geq 2$ vertices. Prove that if $δ(G) \geq \frac{n}{2}$, then $G$ is ...
2
votes
1answer
18 views

Induction proof for a summation

Prove by induction: $\sum_{i=1}^n i^3 = \left[\sum_{i=1}^n i\right]^2$. Hint: Use $k(k+1)^2 = 2(k+1)\sum i$. Basis: $n = 1$ $\sum_{i=1}^1 i^3 = \left[\sum_{i=1}^1 i\right]^2 \to 1^3 = 1^2 \to 1 = 1$. ...
0
votes
1answer
21 views

Sequential criterion for functional limits proof in the opposite direction

Let $f: A\to\mathbb{R}$ Given $c$ is a cluster point in $A$. Prove that the following statements are equivalent: (a) The function $f$ does not have a limit at $c$. (b) There exists a sequence ...
1
vote
1answer
15 views

Understanding a proof that bounded sequences in $\mathbb{R}^p$ has a convergent subsequence

I'm having trouble concerning the following proof that each bounded sequence in $\mathbb{R}^p$ has a convergent subsequence. We have already established that this is true in $\mathbb{R}$ and this is ...
0
votes
1answer
35 views

Proofing the existence of a non-zero congruence class

Let $m\ge2$ be an integer. Show that if there is an integer $a$ such that $\gcd(a,m)=d\not=1$, then there exists a non-zero congruence class $[x]$ in $\mathbb{Z}_m$, such that $[a]\cdot[x]=[0]$. I ...
2
votes
1answer
28 views

Help formalizing this proof about a continuous, one-one function.

I'm having a bit of trouble getting the language on this proof right, though I think I have the idea correct. I have the function $f\colon D \rightarrow {\bf R}$ where $D = [a,b]$. The function is ...
0
votes
1answer
32 views

Prove that there is a 1-1 correspondence between the set of subgroups of $\mathbb{Z}/N \mathbb{Z}$ and the set of the positive divisors of $N$

Im interested in the above Proof, is because I have the intiuition that it is not true at all, because for example, all the primes have exactly 2 positive divisors 1 an themselves, How Can I prove or ...
0
votes
4answers
34 views

Proof without using induction that a number is divisible by 6

Prove without using induction that all numbers of the form $6|8^n - 2^n$. I need a brush up on subtracting numbers with the same base but different exponent. So far I have $8^n - 2^n = 2^{3n} - ...
0
votes
1answer
33 views

Well defined Functions on Congruence classes

Could someone please confirm my logic or point me in the right direction? Thank you. 1) Is the function $f : [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = [n^2]_p$ well defined? 2) Is the ...
0
votes
2answers
96 views

Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$

I'm currently stumped on this question: Let $a$ and $m$ be integers such that $m\ge1$. Consider the congruence class of $a$, i.e., $[a]$ modulo $m$. Prove that: For all $x\in[a]$, ...
1
vote
2answers
46 views

Proving directly that ($a+b)^3 \equiv a^3 + b^3 \mod 3$

Assuming a and b are integers, I must prove directly that: $$ (a + b)^3 \equiv (a^3 + b^3) \mod 3 $$ First, my peers and I made the mistake of assuming what we are trying to prove and thus failed. ...
1
vote
1answer
37 views

Proving that $a$ is an element of a set $A$

I am supposed to prove that if $a \in \mathbb{Z}$ and $a^2\mid a$, then $a \in \{-1,0,1\}$. If I let $B = \{-1,0,1\}$ and $\overline{B} = \mathbb{Z} \setminus B$, is it sufficient to show that $a ...
1
vote
3answers
25 views

Closed form of a sum

I am trying to derive the closed form of the sum $\sum\limits_{i=2}^n \frac{1}{i(i-1)}$ which Mathematica tells me is $\frac{n-1}{n}$. I am completely baffled on how to arrive at this result. The ...
6
votes
1answer
72 views

Solve $3x^2-y^2=2$ for Integers

If $x$ and $y$ are integers, then solve (using elementary methods) $$3x^2-y^2=2$$ I tried the following If $y$ is even, then $4|y^2$ and hence $2|y^2+2$ (and $4$ doesn't divide it), but ...
0
votes
1answer
46 views

Proof of Functions!

Question: Let $ f\colon Z \to Z $ and $ g\colon Z \to Z $ be two functions. Prove that the following are functions. a. $h(x)\colon Z \to Z $ defined as $h(x) = f(g(x))$ when $g(x)$ is an onto ...
1
vote
1answer
37 views

If $a$ divides $b$, then $a$ divides $3b^3-b^2+5b$.

Prove: Suppose $a$ and $b$ are integers. If $a\mid b$, then $a\mid3b^3-b^2+5b$. I think I have an idea of how to prove this, but I'm not entirely sure. I can prove that each individual term in ...
0
votes
0answers
35 views

Well-defined functions using mod $p$ equivalence classes

Prove that if $m, n$ are elements in the set $\mathbb{Z}$ and $m \equiv n \pmod p$, then $m^2 \equiv n^2 \pmod p$. Also, is the function $f: [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = ...
0
votes
1answer
32 views

Equivalence classes of multiples of 3

I'm having a little trouble wrapping my head around the elements of the equivalence classes using the following definition: for m, n in N, define m ~ n if m^2 - n^2 is a multiple of 3. 1) List four ...
0
votes
1answer
26 views

natural deduction on proving a claim

I am working on this proof and wanted to know if I am using the ID natural deduction rule correctly. Can I just assume B and A based on that rule? ...
1
vote
2answers
31 views

finding numbers to make both sides equal

Call a triple-x number an integer $k$ such that $k=x(x+1)(x+2)$ where $x \in Z$. How many triple-x numbers are there between 0 and 100,000? I thought by doing $8!$ and $9!$ would work to see how ...
3
votes
2answers
69 views

How to derive an proof for this infinite square root equation?

Here is continuous square root, namely: $\sqrt {1 + a \sqrt {1+b \sqrt {1+c\sqrt {1 +...}}}}$= any integer Find $a,b,c,d,e,f,...$ in general Uh, very interesting algebra pre-calculus problem, yet ...
0
votes
1answer
25 views

Equivalence Relation and functions question

Could someone please confirm if I understand this correctly? Here is the problem: define ~ on Z by m ~ n in case m^2 ~ n^2. 1) What, if anything, is wrong with the following "definition" of a ...
1
vote
1answer
17 views

Proof of unique solution to a minimization over two sequences

Given two non-strictly ascending sequences, prove that no rearrangement of terms in either sequences will produce a smaller $S$. $A=1,2,3,4,5...\\B=2,3,4,5,6...$ $S=\left|A_1 - B_1\right|+\ldots ...
3
votes
1answer
36 views

Verification of Proof that if f(x) is continuous and periodic then it is uniformly continuous on the reals.

Suppose f is defined on all reals. Then there is a positive p s.t. f(x+p)=f(x) for all x. This is my proof: Assume f is continuous on [0,p] then it is uniformly continuous on [-p,p]. Then for x,y ...
0
votes
2answers
47 views

Proof by induction failure if assumption is wrong?

I never got a clear answer to this question in college. What happens in an induction proof if the assumption is wrong? For example, suppose we try to prove that $n^5$ > n! for n >= $2$ so we start ...
0
votes
2answers
55 views

Can Cantor's theorem prove that N is uncountable (paradox)?

I am struggling a bit trying to understand Cantor's theorem about the reals being uncountable. How can you choose a real number that is different from all real numbers in an enumeration $S$? I ...
3
votes
0answers
57 views

Theorem cannot be proven directly

Can we ever prove a theorem cannot be proven directly (i.e. We must use contrapositive or prove by contradiction.)? Can we even rigorously defined whether a proof is direct or not? Example: I was ...
0
votes
1answer
14 views

Not unit-speed curve on a sphere

This question has already been asked: Curve on a sphere but is slightly different because I don´t have the hypothesis that $\alpha(t)$ is a unit-speed curve; anyway I wanted to do it by myself ...
0
votes
2answers
48 views

How to prove that the diameter of a graph is less than 2 given that the minimum degree of any vertex in G is greater than the number of vertices / 2

How do I prove the following statement. Let $G = (V,E)$ be a graph. Prove that if $δ(G) \ge \frac{|V|}2$, then $\operatorname{diam}(G) \le 2$ I believe $\delta$ is minimum degree of any vertex ...
2
votes
1answer
46 views

What does it mean for a theorem to be “almost surely true”, in a probabilistic sense? (Note: Not referring to “the probabilistic method”)

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them ...
-1
votes
1answer
64 views

Knowing People Proof [on hold]

If I choose any four students among a class, at least one of the four knows all of the other three. Prove that there must be a student who knows everybody in the class.
1
vote
2answers
51 views

Prove this limit of $x^4 + 1/x$ formally

prove: $lim_{x\to 1} \space \space \space x^4 + \frac{1}{x} $ So, $lim_{x\to 1} \space \space \space x^4 + \frac{1}{x} = lim_{x\to 1} \space \space \space x^4 + lim_{x\to 1} \space \space \space ...
0
votes
1answer
17 views

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$ I want to do the $n - 1 \rightarrow n$ induction step. But I'm confused as to what my base case is. Usually if I want ...