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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-1
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5answers
84 views

Prove that the sequence $\sin\left(\frac{n\pi}{3}\right)$ diverges

I don't want to hear that since $sin$ is a periodic function, etc, then we are done. I would like to see a simple proof that make use of the definition of convergence of a sequence. I have tried to ...
1
vote
1answer
23 views

Examples on how to give a proof or a counterexample of a statement

Examples; Prove or give a counterexample of the following statements,with quantifiers: 1) For each non-negative number s, there exists a non-negative number t such that s≥t 2) For each non-negative ...
4
votes
1answer
91 views

Existence of bijection that reorders elements?

Suppose I have some function $f:\mathbb{R}\to[0,1]$. Does there necessarily exist a bijective mapping $g:\mathbb{R}\to\mathbb{R}$ such that $g(x)\leq g(y)$ implies $f(x)≤f(y)$? If not, does it help if ...
1
vote
2answers
27 views

how to prove the uniqueness and existence of equations

I've the equation $e^x=5$, know it has the solution $x=\ln 5$. How to prove the existence before, and after the uniqueness of this solution?
0
votes
2answers
25 views

Discrete Math Combinatorics, permutation, one-to-one proof

I am having trouble getting started with the following proof: (This is homework, so I'd appreciate a nudge in the right direction.) Let m, r $\in$ N with 0 $\leq$ r $\leq$ m. Prove that the number of ...
1
vote
3answers
16 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
0
votes
2answers
47 views

If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$

Just wanted to confirm that this is a correct solution: Proof: Suppose $f(x_0) > 0$ for some $x_0 \in [a,b]$. Then, by continuity of $f$, for $\epsilon < f(x_0)$, there exists $\delta > 0$ ...
0
votes
0answers
41 views

Conjugacy Class Equation

$\newcommand{\cl}{\operatorname{cl}}$Let $p$ be a prime and let $G$ be a group with $|G|=p^n$. Show that $Z(G)\neq\{e\}$. The class equation states $|G| = |Z(G)| + |\cl(a_1)|+\cdots+|\cl(a_n)|$ where ...
2
votes
7answers
188 views

Which number is bigger: $\sqrt[10]{2}$ or $1.2$?

What is the general method for finding such inequalities? I have some more problems of this kind in the text I am using.
0
votes
1answer
23 views

Sequential Criterion for Continuity

I want to prove the following: Let $A$ be a nonempty subset of $\mathbb{R}$, $c\in A$, and $f : A \to \mathbb{R}$. Then $f$ is continuous at $c$ if and only if for every sequence $(x_n)$ in $A$ such ...
6
votes
2answers
67 views

Prove that $ (\mathbb{Q}\times\mathbb{R})\cup (\mathbb{R}\times\mathbb{Q})$ is a locally connected and connected subspace of $\mathbb{R}^2$

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that $A = (\mathbb{Q}\times\mathbb{R})\cup (\mathbb{R}\times\mathbb{Q})$ is a locally ...
0
votes
0answers
6 views

What does $\rho_A\left(\frac{1}{t}\right)$ mean, if the function is the characteristic polynomial?

Given $A=\mathbb{R}^{n\times n}$ and its characteristic polynomial $\rho_A(t)$ show that $q(A^{-1})=0$ where the function is defined as: $$q(t)=\frac{1}{\rho_A(0)}t^n\rho_A\left(\frac{1}{t}\right)$$ ...
0
votes
0answers
26 views

Deciphering whether a set relying on a predicate exists

I'm having some trouble with these style of questions in my Set Theory course and I'm not sure how to proceed with them. Screenshot of the question For part i) I tried something along the lines of ...
2
votes
2answers
40 views

Prove that if $\gcd(m,n)=1$ then every divisor $d|mn$ has a unique form $d=ab$ such that $a|n$ and $b|m$.

I can see why this is true. I have a problem with formality or with explaining certain things properly. An attempt: suppose there are two forms $d=a_na_m=b_nb_m$ such that $a_n,b_n|n,a_m,b_m|m$ but ...
1
vote
0answers
37 views

Product over real interval? Is there a better way of putting this?

In my amateur interest, I have arrived at this (nothing rigorous here at all):$$\prod_{a\in [1,2]} \prod_{b=0}^\infty f(a,b) \neq 0$$ For starters, there might be a more intuitive way about doing ...
1
vote
3answers
67 views

Prove that a subset is normal

Let $H$ be the subset of $GL(2, \mathbb R)$ consisting of all matrices of the form $$ \left[ \begin{array}{cc} x & 0\\ 0 & x\\ \end{array} \right] $$ where $x \neq 0$ Prove that H ...
4
votes
5answers
266 views

Show that $g(a) = g(b) = 0,\ \int_a^b f(x)g(x)dx=0 $ implies $f(x)=0$

Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, I want to ...
0
votes
1answer
22 views

$\sigma$-algebra with cardinality $\aleph_0$ [duplicate]

Can a $\sigma$-algebra in a set $X$ have cardinality $\aleph_0$, the cardinality of the naturals? I do not have a clue on how to start with this? Can someone please give me a hint?
0
votes
3answers
62 views

Beginner linear algebra proof

Hello I am a bit confused on trying to prove a result in linear algebra. I will most what I know but I think it is very incomplete etc. $\mathbf{Thereom}:$ Let $A \in \mathbb{M_{nxn}}$, and v an ...
2
votes
3answers
34 views

If $f$ is a uniformly continuous function show that $g = f(x) - f(y)$ is uniformly continuous on all of $\mathbb{R}^2$

Problem statement: Let $f: \mathbb{R} \to \mathbb{R}$ be a uniformly continuous function on $\mathbb{R}$ and let $g: \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x) - f(y)$. Then $g$ is uniformly ...
1
vote
0answers
30 views

help me to prove this

I would like to prove that $n/k!$ for $k$ that holds $k\leq \frac{\log n}{\log\log n}$ will always be bigger/equal to $1/2$. I tried to use stirling but got stuck. Any ideas? Thanks, Jonatan
2
votes
2answers
37 views

Lebesgue outer Measure of a face of rectangle in $\Bbb R^{n}$

Show that the outer measure of a face $I_1 \times \dots \times I_{i-1} \times \{a\} \times I_{i+1} \times \dots \times I_n$ of a rectangle $I_1 \times \dots \times I_n \subset \Bbb R^{n}$ is zero. ...
1
vote
3answers
22 views

Set Theory Proof with Complements

If $A \cap B = \emptyset$ then $A \subset B'$ and $B \subset A'$, where the prime symbol denotes the complement of each set. Here are my thoughts: Assume $A \cap B = \emptyset,$ since the ...
1
vote
2answers
38 views

Faithful Group Actions and Normal Subgroups

Let $G\curvearrowright X$. Show that $K=\{g\in G:g\cdot x=x,\text{ for all }x\in X \}\trianglelefteq G$. If $\phi\colon G\to Sym(X)$ is the homomorphism given by the action, show that $K=\ker(\phi)$. ...
0
votes
0answers
42 views

What is the sum of reciprocals of Natural Numbers? [duplicate]

I want to calculate the sum of first $n$ natural numbers. I used the following C program to compute the first '$n$' digits : ...
5
votes
3answers
174 views

If $\int_a^b f(x)\,dx=0$, prove that $f(c)=0$ for at least one $c$ in $[a,b]$

Assume $f$ is continuous on $[a,b]$, if $\int_a^b f(x)\,dx=0$, prove that $f(c)=0$ for at least one $c$ in $[a,b]$. The problem didn't state anything about the function $f$, is it safe to assume ...
1
vote
1answer
50 views

Is there any case where a function is 1-1 where the integral=0 and the function is not odd?

I know that the integral of an odd function from -a to a is always equal to 0. Is this only true for odd 1-1 functions? If I know a function is 1-1 and its integral from -a to a is 0, does that imply ...
1
vote
1answer
21 views

Prove $(2^{n}+1)=Q(2^{m}+1)+2^{r}+1$ if r is even and $n=qm+r$, $r<m$

Prove $(2^{n}+1)=Q(2^{m}+1)+2^{r}+1$ if r is even and $n=qm+r$, $r<m$
1
vote
1answer
39 views

Proof by Strong Induction

$a_0 = 1, a_1 = 1, a_k = 2a_{k-1} + 2a_{k_2}$ for $k≥2$ For all integers $n≥0$, $a_n= \frac{1}2[3^{n}+(-1)^n$] Proof By Strong Induction: Basis: $F(0), F(1), F(2), F(3), F(4), F(5)$ Inductive ...
4
votes
3answers
46 views

Proof by Strong Induction for $a_k = 2~a_{k-1} + 3~a_{k-2}$

$$\begin{align} a_0 &= 1 \\ a_1 &= 1 \\ a_k &= 2~a_{k-1} + 3~a_{k-2} \quad \text{ for } k \ge 2 \end{align}$$ Proof by Strong Induction: For all non-negative integers $n$, $a_n$ is an ...
0
votes
1answer
39 views

Prove $a_n = 7a_{n-2} + 6a_{n-3}$ for $n\ge 3$

Let $a_0 = 1$, $a_1 = 1$, and $a_k = 2a_{k-1} + 3a_{k-2}$ for all integers $k\ge 2$. Prove $a_n = 7a_{n-2} + 6a_{n-3}$ for all $n\ge 3$. Proof: Let $n\ge 3$ be arbitrary and fixed. From here, I ...
2
votes
2answers
60 views

Proof of irrationality without using contradiction

I'm just wondering if there exists proofs that certain numbers are irrational that do not begin by saying some like along the lines of "assume $k=a/b$ for integers $a$ and $b$" and then deduce a ...
1
vote
2answers
46 views

How to differentiate $y=(x+1)^3/x^{3/2}$ and $y=2x^4/(b^2-x^2)$

I need to solve a list of derivatives to help me on an exam; however, I'm in doubt when they use another variable (constant) or when I have a fraction with functions that use the power rule. For ...
2
votes
1answer
42 views

Showing $\frac{1}{2\pi i}\int_{\vert \gamma \vert =R}f(\zeta) \frac{1-\frac{Q(z)}{Q(\zeta)}}{\zeta - z}d\zeta$ is a Polynomial

Let $z_1,\ldots, z_n$ be distinct complex numbers contained in the disk $\vert z\vert <R$. Let $f$ be analytic on the closed disk $\vert z \vert \le R$. Let $Q(z)=(z-z_1)\cdots (z-z_n).$ Prove ...
0
votes
2answers
46 views

Do I have the right start for this proof?

I'm trying to prove the following, Suppose R is a partial order on $A$, $B\subseteq A$, and $b\in B$. Prove that if $b$ is the smallest element of $B$, then it is also the greatest lower ...
2
votes
2answers
88 views

Proving $2^{2^n}+3^{2^n}+5^{2^n}$ is divisible by $19$ for all $n\geq 1$ by induction

I came across the following in the book Handbook of Mathematical Induction: $$ 19\mid (2^{2^n}+3^{2^n}+5^{2^n}),\quad n\in\mathbb{Z^+}\tag{1} $$ Apparently, this problem is not so bad if you think ...
0
votes
2answers
72 views

Proof that Electromagnetic force is much stronger than gravity [closed]

The electromagnetic force is given by this formula: $$Fl=q(v\cdot\,B)$$ and gravity is given by $$Fz=m\cdot\,g$$ Now do I've to proof that the electrmagnetic force is much stronger than gravity but ...
0
votes
1answer
56 views

Proof using axioms of real numbers

Working on proof writing, and I need to prove $$(-x)y=-(xy)$$ using the axioms of the real numbers. I know that this is equivalent to saying that the additive inverse of $xy$ is $(-x)y$ but I am ...
0
votes
1answer
36 views

Proving a linear transform defined by an integral is injective

Let the fact that $I(p)(x)=\int_0^x p(s) ds$ is a linear transform from $P_4\rightarrow P_5$ be given. Prove that $I$ is injective. Would it be sufficient to just state that for any 2 ...
0
votes
2answers
32 views

Order of logical quantifiers within a statement

I understand that the order of the quantifiers of a statement determine the truth value of statement. For example, $$\forall x \in \mathbb{R}, \exists y \in \mathbb{R}\ \text{such that}\ ...
1
vote
1answer
61 views

Bounding the edges belonging to no perfect matching

We are told to let $G = (X \cup Y, E)$ be a bipartite graph with $|X|=|Y|=n$, and to suppose that $G$ has a perfect matching. I am trying to find a way to prove that $G$ has at most $n \choose 2$ ...
0
votes
1answer
36 views

Proving a linear transform is injective

Let $A:V \rightarrow W$ be a linear map. Prove that A is injective iff $\{v \in V :Av=0\}=\{0\}$ I read that a linear transform is injective iff the kernal of the function ...
0
votes
0answers
31 views

Proving greatest lower bound and least upper bound.

Could someone tell me how I can define glb and lub with logical/set symbols for the following question? I don't need complete proofs, I would like to know the givens and goals to work with here. ...
1
vote
3answers
47 views

Prove by induction $n^2 \leq n!$ for $n\geq 4$.

I managed to get $P(4):4^2 = 16 \geq 24 = 4!$ But then assuming $n^2 \geq n!, \forall n\geq4\in\mathbb{Z}$, I need to prove $(n+1)^2 \geq (n+1)!$ I tried $n^2+2n+1\geq n!\cdot (n+1)$, but I got ...
2
votes
1answer
36 views

Proof of linear independent eigenvectors

Hello I am looking for insight onto this theorem I will post. I will also post what I have for a potential proof, but I don't think it is very rigours. I am looking for one that maybe uses induction ...
1
vote
2answers
23 views

Prove that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses

Prove that if $f$ is a function such that $f'(x) > 0$ $\forall x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then $f$ ...
0
votes
2answers
46 views

How do I prove this? (Relations Proof)

So I can't seem to figure out how to prove this. Any help would be greatly appreciated. My professor said a contradiction would work but I don't see where I can make a contradiction. Show that {X ...
1
vote
3answers
32 views

Disproving a inequality implication by contradiction.

Let $x,y \in R$. If $0 \leq y < x$ for all $x > 0$, then $y=0$. Proof by contradiction: Assume the opposite that is; "If $0 \leq y < x$ for all $x > 0$, then $y\neq0$". ...
0
votes
1answer
11 views

Proving an existential quantified goal with a single universal quantified given

I'm trying to prove the following with no success: $\forall$A $\in$ F $\forall$B $\in$ G(A $\not\subseteq$ B) $\leftrightarrow$ $\cup$F $\not\subseteq$ $\cup$G In order to prove this statement, I ...
3
votes
4answers
66 views

Proving $ax+b$ is a linear function

$L\colon\mathbf{R}\to \mathbf{R}$ be given by $L(x)=ax+b$ over the scalar field $R$. I understand for that a function to be linear, it must adhere to the properties of additivity and scalar ...