For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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2
votes
3answers
49 views

Is this a valid way to prove that $\frac{d}{dx}e^x=e^x$?

$$e^x= 1+x/1!+x^2/2!+x^3/3!+x^4/4!\cdots$$ $$\frac{d}{dx}e^x= \frac{d}{dx}1+\frac{d}{dx}x+\frac{d}{dx}x^2/2!+\frac{d}{dx}x^3/3!+\frac{d}{dx}x^4/4!+\cdots$$ ...
0
votes
3answers
33 views

Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)

Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3) SO this is for abstract algebra and I am really struggling with this. Here are some of ...
0
votes
0answers
19 views

My proof that there are primitive roots modulo $p^2$

Let $p$ be a prime number. I'd like to prove that there are primitive roots modulo $p^2$. Could someone check this argument? Note that if $r\in\mathbb Z$ is a primitive root modulo $p^2$, it must ...
1
vote
1answer
12 views

Proof by induction that for a complete simple and undirected graph that $|E|=\frac {n(n-1)} 2$

Prove with induction that for a complete simple and undirected graph that $|E|=\frac {n(n-1)} 2$ Base case is trivial. Suppose that for a graph with $n-1$ vertices we have $|E|=\frac ...
3
votes
1answer
35 views

Axler LADR Exercise

The exercise is: Suppose $v_1, \ldots , v_m$ is linearly independent in $V$ and $w \in V$. Prove that if $v_1+w, \ldots, v_m+w$ is linearly dependent, then $w \in \operatorname{span}(v_1, \ldots, ...
2
votes
1answer
21 views

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational.

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. Take ...
-1
votes
1answer
15 views

question on proving inequalities [on hold]

If I need to prove $t(x) \ge0 $, for all $ x>0$ and I prove that $t(x) \gt 0 $, for all $ x>0$ does that make for a proof or is it wrong?
1
vote
1answer
30 views

Showing that the class of all sets of a particular cardinality is not a set.

How to show that the class of all sets of a particular cardinality ,say $h$ is not a set. My argument: I assume that I've shown the following lemma. Lemma: If $X$ is an infinite set of cardinality ...
1
vote
1answer
34 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
-3
votes
0answers
32 views

Prove the inequality between the arithmetic and geometric mean

Assume that for $x_1,...,x_n\geq0$ we let $G=(x_1x_2\dots x_n)^{1/n}$ and $A=(x_1+x_2+...+x_n)/n$. I would like to know if the following procedure leads to a proof of $$G\leq A$$ The equality is ...
0
votes
1answer
38 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
0
votes
0answers
33 views

Inverse property for groups Proof

I was wondering if (1) this proof is correct, and (2) if other proofs exist for the following: Prove that $(a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1}$ where $a_i \in $ a Group $G$ Proof by ...
0
votes
2answers
21 views

Help with proof about functions and subsets

Problem: let $f: A \rightarrow B$. Prove that $f$ is injective if and only if for all $D \subset A$ we have that $f^{-1}(f(D)) = D$. Proof: => Suppose $f$ is injective. Let $x \in f^{-1}(f(D))$. ...
1
vote
0answers
49 views

Is my understanding of the argument correct?

I worked through a proof of: $$ f(z) = {1\over 2 \pi i}\int_{\partial D} {f(w) \over w -z} dw$$ where $D\subset \mathbb C$ is an open disk and $f$ is holomorphic on $D$ and continuous on ...
1
vote
1answer
26 views

Proof of the second principle of mathematical induction

This was an exercise in my lecture notes for which no answer was provided, so I seek verification on whether my proof is correct. Prove that if 1. $P(n_0)$ is true for some $n_0 \in \mathbb N$, and ...
0
votes
0answers
21 views

Show that $\int_a^b f(x) dx=\lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} \int_{x_k}^{x_{k+1}} f(x) dx$.

I've come up with a proof for the following statement, but I'm not quite sure it's 100% correct. I would appreciate any help: If $f$ is integrable on $[a,b]$, $x_0=a$, and $x_n$ is a sequence of ...
1
vote
2answers
65 views

$V$ is finite dimentional over field $K\iff$ field extension $L/K$ is finite

Let $L/K$ be a field extension and $V$ a non-zero vector space over $L$. Prove that: $V$ is finite dimensional over $K\iff V$ is finite dimensional over $L$ and $[L:K]<\infty$ for the first ...
0
votes
0answers
18 views

Conditional independence given elementary events implies conditional independence given $\sigma $-algebra

Proposition: Let $X$ be a continuous markov chain with discrete state space $S$. Let $Z$ be the corresponding jump chain and $\left\{ {{W_i},i \in \mathbb{N}} \right\}$ its holding times. Let ...
0
votes
3answers
39 views

Help me with proof concerning functions

Problem: Let $X$ and $Y$ be non-empty sets and let $f: X \rightarrow Y$ be a function. We define $F: P(Y) \rightarrow P(X)$ by $F(B) = f^{-1}(B)$ for all $B \in P(Y)$. Proof that $F$ is injective if ...
1
vote
0answers
23 views

Using Intermediate value theorem and Rolle's theorem

Find how many solutions $2\ln x+2x^2+7=0$ has. Define: $f(x)=2\ln x+2x^2+7$, derive it and equate to $0$: $f'(x)=0 \\ 2+4x^2=0$ The discriminant is negative so there are no solutions, so from ...
0
votes
1answer
28 views

Riemann Integral Property for Continuous, Monotonic, Non-negative Function

If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$ My attempt: Define $F(x)=\int^{x}_{0} f(t)dt$. ...
0
votes
1answer
29 views

Given the distribution of $X$ and $Y=-2\theta \ln X$. How is $Y$ distributed?

The pdf of $X$ is $f(x) = \theta x^{\theta-1},\enspace 0<x<1, \enspace 0<\theta<\infty.$ Let $Y=-2\theta \ln X.$ How is $Y$ distributed? My work: $$ \begin{align*} F(Y) = P(Y \leq y) ...
4
votes
0answers
29 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
2
votes
3answers
56 views

prove or disprove (discrete math)

This the question: Q: Prove or disprove the following statement. The difference of the square of any two consecutive integers is odd This is working step: let $m,m+1$ be 2 consective ...
1
vote
2answers
65 views

Is it true that $a$ can't be zero in the quadratic function $y=ax^2+bx+c$?

I read that for $y=ax^2+bx+c$ is a quadratic function where $a\neq0$, but is it true that $a$ really can't be zero? I think it is because if $a$ was zero, there wouldn't be a parabola. There would ...
2
votes
1answer
39 views

How to adapt proof by contradiction showing that a sqrt(2) is irrational for sqrt(20)?

This example is from Discrete Math and its Applications I understand the steps the author is taking. First he assumes sqrt(2) is rational meaning that there exists integers a, and b such that ...
0
votes
1answer
28 views

Let $G$ be a connected graph, then $G$ is a tree iff $G$ has no cycles

Prove the following: Let $G$ be a connected graph, then $G$ is a tree $\iff$ $G$ has no cycles. $\Rightarrow$ If $G$ is connected and a tree then by the definition of tree it has no cycles. ...
2
votes
2answers
34 views

Integration by parts and $dx$ notation

Please overview this integral evaluation: $$ \int x^3 \arctan(x^2)dx = \frac{x^4}{4}\arctan(x^2) - \int \frac{1}{1+x^4}2x dx $$ Let's evaluate the right term: $$\int \frac{1}{1+x^4}\color{Blue}{2x ...
0
votes
1answer
9 views

Proof concerning indexed family of sets

Let $f: A \rightarrow B$ be a function. Let $I$ be a non-empty set, and let $\left\{U_i\right\}_{i \in I}$ be a family of sets indexed by $I$ such that $U_i \subset A$ for all $ i \in I$. Proof the ...
2
votes
1answer
33 views

Asymptotics of $\sum_{n\leq x}\tau_{k}\left(n\right)$

We define $\tau_{k}\left(n\right)$ to be the number of ordered $k$-tuples of positive integers with product equal to $n$. It is easily shown that this satisfies the recurrence relation ...
0
votes
1answer
26 views

$\lim_{x\to x_0 ;x\in X} f ( x)$ exists if f is a uniform continuous function and $x_0$ is an adherent point

Proposition: Let $X$ be a subset of $R$, let $f:X\to R$ be a uniformly continuous function, and let $x_0$ be an adherent point of $X$. Then $\lim_{x\to x_0 ;x\in X} f ( x)$ exists. Proof Take any ...
2
votes
1answer
22 views

Given a graph on $n$ vertices find the maximum amount of edges so it can be colored with no monochromatic $K_m$

I invented a problem and I wanted to share :What is the maximum amount of edges a graph on $n$ vertices can have if it can be edge-colored with $k$ colors so that it does not have a monochromatic ...
1
vote
1answer
28 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
0
votes
1answer
13 views

Prove every integral ideal $J$ is identical with $\Bbb{J}_m$ for some $m$.

Prove every integral ideal $J$ is identical with $\Bbb{J}_m$ for some $m$. Suppose $J \neq \{0\} = \Bbb{J}_0$. By the least integer principle, there exists an $m \in J$ such that $rm \in J$ in $r ...
2
votes
2answers
221 views

Is this logically valid?

$$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1} > ln(n)$$ and so, necessarily, $$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1}+\frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n} > ln(n)$$ ...
0
votes
0answers
16 views

Convolution of negative binomial distribution w/ generalized binomial theorem

This is Exercise 3.1.1 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Show that $b^−_{r,p} \ast b^−_{s,p} = b^−_{r+s,p}$ for $r, s \in (0,\infty)$ and $p \in (0,1]$. ...
0
votes
1answer
19 views

Use the least integer principle to prove the following.

Least integer principle: Every non-empty set of positive integers has a least element. Using this fact, define $r$ to be the least integer for which $j - qk > 0$ where $j, k \in \Bbb{Z}$ ...
0
votes
1answer
14 views

Proving inverse implication by conversion

I have proven logically that the inverse of an implication is true if and only if the converse of said implication is true (as shown below). proposition 1: k has same parity as 2j proposition 2: k ...
0
votes
0answers
13 views

Prove for each pair of integers $j, k : k > 0$, there exists a $q : j - qk > 0$

Prove for each pair of integers $j, k : k > 0$, there exists a $q : j - qk > 0$. I began by writing out all three cases, i.e. $C_1 \to j > k$, $C_2 \to j = k$, and $C_3 \to j < k$. ...
1
vote
0answers
22 views

Does this reasoning about fourier analysis make sense?

I'm asked to show that there cannot be $\alpha_1,\alpha_2,...\in\mathbb{C}$ s.t. $$\lim_{N\to\infty}\int_{-\pi}^{\pi}|e^{it}-\sum_{k=1}^{N}a_k\sin(kt)|^2dt=0$$ Here is my attempt: Assume there are ...
0
votes
1answer
38 views

prove $\lim_{x\to 0} f(x) = 0$ using epsilon delta

prove using Epsilon Delta that $\lim_{x\to 0} f(x) = 0$, where $f(x) = \left\{ \begin{array}{l l} \;\;\; \sqrt6 \;x & \quad \text{if } {x \in \mathbb{Q}}\\ -\sqrt6 \;x & \quad ...
0
votes
1answer
27 views

Integral of nonnegative function on plane domain gives a negative result, what is wrong?

Given an area $D: x \ge y, 0 \le x \le 1, y \ge 0$. $$ f(x,y)= \begin{cases} 2, & (x,y) \in D,\\ 0, & \text {others}\end{cases} $$ For this area $D_1: x+y \le 1, 0 \le y \le x$, I'm ...
1
vote
4answers
57 views

Prove that if $a<b$, then $-b<-a$

Prove that if $a<b$, then $-b<-a$ I'm a bit lost in this one, this is what I did: First case: $0<a<b$ $|a|<|b|$, so $|-a|<|-b|$ Since both are negative and $|-b|$ is greater ...
2
votes
2answers
36 views

Is $\sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous?

$f(x) = \sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous because: Proof: $f(x)$ is a continuous function on $[0, 1]$, which is a closed interval, so $f$ is uniformly continuous on ...
1
vote
1answer
35 views

Clarification on Cantor Diagonalization argument?

My book is Discrete Mathematics and its Applications. This is its section on Cantor's Diagonalization argument I understand the beginning of the method. The author is using a proof by ...
2
votes
3answers
56 views

How to approach this proof problem, what proof to use, what assumption to use?

This is a problem from Discrete Mathematics and its Applications Here is the definition of rational that my book uses Usually when I approach this type of a problem, I can find a type of proof to ...
0
votes
0answers
20 views

Would it be necessary to have another proof within the proof by cases in this problem?

This is a problem from Discrete Mathematics and its Applications I am using Proof by Cases. This is my book's definition on it. Here is my work so far I tried to leverage without of generality ...
1
vote
0answers
32 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
2
votes
2answers
42 views

How can we show that “almost surely” equal random variables have the same distribution?

How can we show that "almost surely" equal random variables have the same distribution? We know $X =\text{(a.s)} Y$. What I have so far: $$\begin{align*}\implies& P(X = Y) = 1 \\ ...
0
votes
1answer
44 views

Proof $\sum{ k{ x }^{ -k }=\frac { x }{ { (x-1) }^{ 2 } } }$

As the title says, I want to prove the following: $$\sum {k{x}^{-k}=\frac{x}{{(x-1)}^{2}}}$$ But I think I am doing something wrong. I start from the following: $$\sum{x^k} = \frac{x}{1-x} \implies ...