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

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1
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4answers
80 views

More limits and derivatives, can I do this?

Here is the given problem: Suppose $f(0)=0$ and $f'(0)=-1$. Evaluate the following limit if it exists: $$\lim\limits_{h \to 0} \frac{f(h)-f(2h)+f(3h)-f(4h)+...+f(2013h)}{h}$$ So what I was thinking ...
1
vote
2answers
74 views

Proving $\lim_{n \to\infty} \frac{1}{n^p}=0$ for $p > 0$?

I'm trying to prove 3.20a) from baby Rudin. We are dealing with sequences of real numbers. Theorem. $$\lim_{n \to {\infty}} \frac{1}{n^p} = 0; \hspace{30 pt}\mbox {$p > 0$}$$ Proof. ...
1
vote
1answer
68 views

Proving (by using Zorn's lemma) that every nonempty set contains a maximal ideal

I am trying to prove the following exercise: Let $X \neq \emptyset$. Prove, (by using Zorn's Lemma) that there exists a maximal ideal in $P(X)$. Proof: Take $\mathcal{J}$ to be the set of all ideals ...
2
votes
1answer
60 views

Limit of a sum of roots proof

Given the sequence: $$a_n=\alpha\sqrt{n+a}+\beta\sqrt{n+b}\ with\ \ \alpha,\beta,a,b\in\mathbb{R}\ and\ \alpha,\beta\neq0$$ Prove that $$\lim_{ n\to \infty} a_n = 0\ iff\ \alpha=-\beta$$ I start ...
3
votes
0answers
83 views

Why is this proof false? (Why is $e^i \neq 1$?) [duplicate]

I found this on MathOverflow: $$e^i = (e^i)^{(2\pi/2\pi)} = (e^{2\pi i})^{1/2\pi} = 1^{1/2\pi} = > 1.$$ I first saw this one many years ago, written on the wall of a bathroom stall in ...
1
vote
0answers
111 views

alternate proof of theorem 1.21 of Baby Rudin

I wanted to ask if anyone has tried to prove theorem 1.21 from Baby Rudin book (on existence and uniqueness of positive real n-th root of a positive real number) differently and would care to check my ...
1
vote
1answer
90 views

Review of solution: Prove $\liminf({a_n}) \ge \liminf({b_n})$

${a_n} \ge {b_n}\forall n \in $ Prove: $\liminf({a_n}) \ge \liminf({b_n})$ I proved it by contradiction. Let's assume $\liminf({a_n}) < \liminf({b_n})$. $a := \liminf({a_n})$ $b := \liminf({b_n})$ ...
0
votes
3answers
337 views

Line and a triangle never intersect at exactly 3 points - proof verification

Let´s suppose that there is a line l on which lie exactly 3 points of some triangle. Lets assume that none of the points are vertices of the triangle. Then, since no line intersects a side of a ...
1
vote
2answers
135 views

Determine the trigonometric Fourier series

Consider the function $$ f(x):=\begin{cases}x(\pi-x), & x\in [0,\pi]\\-x(\pi +x), & x\in [-\pi,0]\end{cases} $$ and calculate its trigonometric Fourier series. Hello! So ...
1
vote
1answer
29 views

Existence of isomorphism between $\mathbb{Z}/(p)$ and some finite group $G.$

Let $G$ be a group with $|G|>1.$ Suppose $G$ and $\{e_G\}$are only subgroups of $G.$ Then, there exists $p \in \mathbb{P}$ such that $G \cong \mathbb{Z}/(p).$ May I know if my proof is correct? ...
0
votes
2answers
113 views

Let $G$ be a group of order $27.$ If $G$ is not abelian, then $|Z(G)|=3,$ where $Z(G) = \{z \in G: zg=gz, \forall g \in G\}$

Claim: Let $G$ be a group of order $27.$ If $G$ is not abelian, then $|Z(G)|=3,$ where $Z(G) = \{z \in G: zg=gz, \forall g \in G\}$ May I know if my proof is correct? Thank you. Lemma: $G/Z(G) $ is ...
2
votes
2answers
356 views

Transpose of an invertible linear transformation..

I am trying to prove that suppose that a linear transformation $T$ is invertible, then its transpose $T^t$ is also invertible. Is the following proof correct? Proof: Let $T$ be an invertible ...
3
votes
1answer
60 views

Limit Proof Check

Reviewing limits and I'm afraid i may be making mistakes, just looking for a quick proof check. $f(x)=x^4$, prove that $\lim _{x \rightarrow a}=a^4$ by showing how to find $\delta$ . This is my ...
0
votes
2answers
39 views

Some problems in group theory

May I know if my proof/solution is correct? Thank you v. much. 1.) If $G, H$ are finite groups of order $10$ and $21$ respectively, then every homomorphism $f:G \to H$ satisfies $f[G] = \{e_H\}.$ ...
3
votes
2answers
508 views

Find a basis for the subspace sum and then calculate its dimension.

By definition, $U + V = \{\mathbf{u} + \mathbf{v} : \mathbf{u} \in U\ \; \& \; \mathbf{v} \in V\}$. Let $U = \{ \; u_1 = (1, 1, 0, \color{green}0), u_2 = (-3, 7, 2, \color{green}1) \;\}, V = ...
0
votes
1answer
416 views

Relatively prime polynomials

If $f(x)$ is relatively prime to $p(x)$ in $F[x]$ prove that there is a polynomial $g(x) \in F[x]$ such that $f(x)g(x) ≡ 1_F \pmod{ p(x)}$. Now it has just occured to me that this is a field we are ...
2
votes
1answer
331 views

If $H,K$ are subgroups of finite group $G,$ then $|G:(H \cap K)|\leq |G:H||G:K|$

If $H,K$ are subgroups of finite group $G,$ then $|G:(H \cap K)|\leq |G:H||G:K|$ May I know if my proof is correct? Thank you v. much. Proof: $$|G:(H \cap K)| \leq |G:H||G:K| ...
1
vote
1answer
75 views

Correctly representing a $2^n < n!$ statement

$$2^n < n!$$ After an inductive proof I determined that $2^n < n!$ is valid only for values greater than or equal to $4$. So. How do I represent this conclusion? Is this correct? $$\forall n ...
2
votes
1answer
121 views

Generalized distributive laws proof feedback

I'm currently learning proofs and elementary set theory. I would like to have feedback on my proof since I'm self-studying. Are some part superfluous or unclear? My proof goes as follows: I will ...
1
vote
3answers
332 views

verify identities $\cos(6x) = 1-2 (2 \sin (x) \cos^2 (x ) + \cos(2x) \sin(x))^2$

i'm trying to verify this identity and i'm kinda stuck. I will appreciate any help! $\cos{6x} = 1-2 (2 \sin {x} \cos^2 {x} + \cos{2x} \sin{x})^2$
1
vote
1answer
243 views

Proving the Mean Value Theorem

Theorem The Mean Value Theorem: Let there be a function $f$ which is continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $f'(c)=\dfrac{f(b)-f(a)}{b-a}$ I'd like to prove the ...
7
votes
2answers
191 views

Let $G$ be a finite group and $\phi:G \to K$ be a surjective homomorphism and $n \in \mathbb{N}. $ If $K$ has an element of order $n,$ so does $G.$

Let $G$ be a finite group$\ ,\phi:G \to K$ be a surjective homomorphism and $n \in \mathbb{N}. $ If $K$ has an element of order $n,$ so does $G.$ May I know if my proof is correct? Thank you. ...
3
votes
0answers
82 views

Proving a case of the Intermediate Value Theorem

I'd like to prove the following lemma: Lemma: Let a function $f:[a,b]\to\mathbb{R}$ be continuous and suppose $f(a)<0<f(b)$. Then, there exists a $c\in(a,b)\ni f(c)=0$. My attempt at the ...
1
vote
0answers
75 views

Proving Riemann integrability of a function.

Define a function $f$ by $$f(x) = \begin{cases}42 & \text{if }x =1,2,3,4; \\0 & \text{otherwise} \end{cases}$$ Prove that $f$ is integrable on $[0,5]$ by using the box-sum criterion. ...
2
votes
0answers
68 views

Proving the converse of the Cauchy criterion for integration

Prove the converse of the Cauchy criterion for integration. That is, prove that if $f$ is integrable on $[a,b]$, then for any $\epsilon>0$ there is a $\delta>0$ so that for any partitions ...
0
votes
1answer
108 views

Sequence of functions involving enumeration of rational numbers.

Let $\{r_1,r_2,...,r_n,...\}$ be an enumeration of the rational numbers in $I:=[0,1]$ and define $f_n:I \to \mathbb{R}$ by $f(x) =1 , $ if $x = r_1,...,r_n$ and $f(x) = 0,$ otherwise. Find ...
5
votes
2answers
127 views

$R$ integral domain implies $R[x]$ integral domain

$R$ integral domain implies $R[x]$ integral domain. MY attempt: Take $f = a_0 + a_1x + \cdots + a_nx^n, g = b_0 + \cdots + b_mx^m \in R[x] $. Suppose $f(x) \neq 0 $ and $a_n \neq 0 $ and say $fg ...
1
vote
0answers
56 views

Subgroup and Normal in Group Theory

Let $N$ be a subgroup of $Z(G)$. Show that $N$ is normal in G. My Proof: $N\leq Z(G)$ (subgroup) and $Z(G)\leq G$ so $N\leq G$. $\forall h\in N$ we have $h\in Z(G)$ and this means $\forall g\in G$ ...
1
vote
1answer
241 views

Uniform convergence of composite functions

Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions that converges uniformly to $f$ on $A$ and $|f_n(x)| \leq M, \forall x\in A, \forall n \in \mathbb{N}.$ If $g$ is continuous on $[-M,M],$ does ...
6
votes
0answers
343 views

bisectors of an angle in a triangle intersect at a single point - proof verification

Let´s consider a general triangle ABC. Let´s draw two angle bisectors from vertices A and B. It is obvious that these two angle bisectors intersect at a single point X. Since X lies on the angle ...
2
votes
2answers
379 views

Proving that Zorn's Lemma implies the axiom of choice

I am trying to solve an exercize, in which we were asked to prove that Zorn's lemma implies axiom of choic. I am using a guidness that was given that said we should use a set $\mathcal{F}$ which i'll ...
0
votes
1answer
87 views

Prove that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\infty}\int_E|f_n|=\int_E|f|.$

I'm reading Real Analysis by Royden 4th Edition. The entire problem statement is: Let $\{f_n\}_{n=1}^\infty$ be a sequence of integrable functions on $E$ for which $f_n\to f$ pointwise a.e. on $E$ ...
9
votes
1answer
103 views

Proving $(2n-1)^n + (2n)^n ≈ (2n+1)^n$

As I do, I was messing around and I thought to myself this simple thing: $3^2 + 4^2 = 5^2$ I just thought that this is only Pythagorean triplet with sequential integers. I know that there are no ...
0
votes
2answers
74 views

Approach Question: Prove that $¬∃n∅ = n^+$

I wonder if I have approached this in the right way. I'm not sure if I have interpreted the question correctly, or made correct use of the successor function. Thank you in advance Question: ...
1
vote
1answer
137 views

Question on Convergence of Improper integrals

Question is to check which of the following improper integrals are convergent? $$\int _1^{\infty} \frac{dx}{\sqrt{x^2+2x+2}}$$ $$\int _0^5 \frac{dx}{x^2-5x+6}$$ $$\int ...
0
votes
3answers
46 views

Prove that if $F: A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is one-to-one

Prove that if $F: A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is one-to-one Proof: Suppose $F$ is not one-to-one. Then there exist $x_{1}, x_{2} \in A$ such that $F(x_{1}) = F(x_{2})$ where ...
2
votes
1answer
29 views

Let $f: A \rightarrow B$, $D \subseteq A$, and $E \subseteq B$. Prove that $f^{-1}(B - E) \subseteq A - f^{-1}(E)$

Let $f: A \rightarrow B$, $D \subseteq A$, and $E \subseteq B$. Prove that $f^{-1}(B - E) \subseteq A - f^{-1}(E)$ Proof: Let $x \in f^{-1}(B-E)$, then $x \in f^{-1}(B)$ and $x \notin f^{-1}(E)$... ...
3
votes
0answers
48 views

Show equivalences concerning independence

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space. We say, that $(E_i\in\mathcal{A}:i\in I)$ is a family of independent events, if for any finite subset $I_0\subset I$ it is ...
2
votes
0answers
49 views

Can anyone explain me this proof about a Brownian Motion?

Prove that the process $W_t=(1+t)U_{t/(1+t)}$ on $[0,\infty)$ is a Brownian motion. $\text{(b)}$ Clearly $Y_0=U_0=0$, and inherits continuity of sample paths from $U_t$ (and hence from $W_t$). ...
3
votes
1answer
39 views

Show: $\mathcal{L}^p\subset\mathcal{L}^q$

Consider a measurable space $(\Omega,\mathcal{A},\mu)$. Let $\mu(\Omega)<\infty$ and $1\leq q\leq p$. Show that then $$ \mathcal{L}_{\mu}^p\subset\mathcal{L}_{\mu}^q. $$ Good ...
1
vote
1answer
64 views

Prove that if $F : A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is Injective

Statement: if $F : A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is $1-1$ Proof: If $F$ is not $1-1$, then there exist $x_{1}, x_{2} \in A$ where $x_{1} \neq x_{2}$ and $F(x_{1}) = F(x_{2})$. ...
2
votes
0answers
44 views

Prove that $f: \mathbb{N} \rightarrow \mathbb{N}-\left \{ 1 \right \}$ given by $f(x) = x+1$ is $1$-$1$ and onto

$f: \mathbb{N} \rightarrow \mathbb{N}-\{1\}$ given by $f(x) = x+1$ is $1$-$1$ and onto. Proof: ($1$-$1$) Suppose $f(x_{1}) = f(x_{2})$ for $x_{1}, x_{2} \in \mathbb{N}$. Then $x_{1} + 1 = x_{2} + ...
0
votes
1answer
83 views

Verification: Proof that the Context-Free Languages are Closed under Reversal

Here's the problem: Prove that the context-free languages are closed under reversal. Here's my work: We want to show that if $L$ is a context-free language, then $L^R$ is a context-free ...
5
votes
3answers
146 views

Can Peirce's Law be proven without contradiction?

Good evening, I heard the proof by contradiction is required for Peirce's law. AFAIK, truth tables are not related directly to proofs by contradiction, and if of an operation $\text {op}$ we have a ...
7
votes
1answer
154 views

Prove that $ \sum_{1 \le t \le n, \ (t, n) = 1} t = \dfrac {n\phi(n)}{2} $

Problem: Prove that the sum of all integers $ t \in \{ 1, 2, \cdots, n \} $ and $ (t, n) = 1 $ is $ \dfrac {1}{2} n \phi (n) $, where $ \phi $ is the Euler Totient Function. My proof: Define the ...
0
votes
3answers
159 views

Help with Pointwise and Uniform Convergence in Metric Spaces

I am having a bit of difficulty understanding uniform convergence and would also like to check my understanding of pointwise convergence. Using the example of $f_n$(x) = $x^n$ on (-1,1), I found the ...
2
votes
1answer
1k views

Prove the proposition: there are infinitely many primes of the form 4k + 3, where k ≥ 0 is an integer

Proposition 2. there are infinitely many primes of the form 4k + 3, where k ≥ 0 is an integer. (a) Let n ∈ N. Suppose q1,q2,...,qn are positive integers such that for all 1 ≤ i ≤ n, each qi = 4ki + ...
1
vote
1answer
117 views

Rudin's proof of the existence of a Haar Measure: why is his functional continuous?

I'll start with some notation: $G$ is a compact group, $H_L(f)$ is the convex hull of the set of left-translates of a function $f:G \rightarrow \mathbb{C}$, $K_f$ is its closure (which is shown to be ...
2
votes
1answer
74 views

Evaluate $\int_{\partial \mathbf{D}} f(z) dz$ for some meromorphic $f$.

This is for homework, so just hints please! The question asks If $f$ is a meromorphic function in $\mathbb{C}$ that satisfies $|f(z) z^2| \leq 1$ for $|z| \geq 1$, then evaluate $\int_{\partial ...
0
votes
1answer
69 views

Why is this proof by induction incorrect? [closed]

Let $P(n)$ be the statement "$\sum_{i=1}^{n}i=\frac{(n+\frac{1}{2})^2}{2}$". Basis Step: Clearly $P(1)$ is true as the formula holds for $n=1$. Inductive Hypothesis: Suppose that $P(k)$ is true for ...