3
votes
1answer
12 views

Meromorphic and even

I would like to do the following exercise : Let $f$ be a meromorphic function and $\mathcal{P}$ the set of its poles. We also assume that $f$ is even ($\forall z \in \mathbb{C}, \; ...
1
vote
1answer
28 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
2
votes
0answers
20 views

Question about integration and sets of measure zero

Let $Q \subseteq \mathbb{R}^n$ be a box, $f: Q \to \mathbb{R}$ be bounded, integrable on $Q$. Suppose $g: Q \to \mathbb{R}$ is another bounded function such that $f(x) = g(x)$ for any $x \in Q ...
0
votes
0answers
37 views

Consecutive natural numbers [duplicate]

Please I want to know what is the most appropriate expression that if it is asked to find the counterexample of "The product of any three consecutive natural numbers is divisible by 9" My expression ...
1
vote
0answers
56 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by producing a linear function

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
1
vote
2answers
61 views

Prove by induction $1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n} \ge 1 + \frac {n}{2}$

Prove by induction $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{2^n} \ge 1 + \frac {n}{2}$ I can't explain in words how the left hand side of the equation is achieved soI shall ...
1
vote
2answers
52 views

$\varepsilon$-$\delta$-definition for continuity of $x^n$

Show that $f:\Bbb R\to\Bbb R,x\mapsto x^n$ with $n\in\Bbb N$ is continuous in $x_0=0$ using the $\varepsilon$-$\delta$-definition. We assume that $$\forall ...
1
vote
0answers
19 views

Continuity of a function on a box $Q$ implies integrability

Let $Q \subseteq \mathbb{R}^n$ be a box, and say $f: Q \to \mathbb{R}$ is continuous, then $f$ is integrable on $Q$. MY ATTEMPT: Since $f$ is continuous on $Q$, then $f$ must be uniformly continuous ...
2
votes
0answers
33 views

Please check these proofs for sets

I would appreciate the insight again for a couple of proofs since I'm learning. These are homework problems in so much as they are problems from the textbook. They are not required by my professor. ...
1
vote
1answer
50 views

Prove thoroughly: If the degree of all vertices is greater or equal to $\frac{|V| - 1}{2}$, then the simple graph is connected.

I am struggling to write a good, thorough proof. The proof is supposed to be logically rigorous, correct and complete (e.g. no hidden assumption). Moreover, style is important - the proof should be ...
3
votes
3answers
87 views

Please check this proof

I'm taking Discrete Math this semester. While I understand the mechanics of proofs, I find that I must refine my understanding of how to work them. To that end, I'm working through some extra ...
0
votes
1answer
22 views

Normal Subgroups proof help

Show that if $H$ is a subgroup of $Z(G)$, then $H$ is a normal subgroup of $G$. This is what I have so far. proof: Suppose $H$ is a subgroup of $Z(G)$. Let $h$ be in $H$ and $g$ in $G$. Then ...
1
vote
2answers
54 views

l'Hopitals rule - is my working correct?

Is anyone able to help me with this question on l'Hopital's rule? Use l'Hopital's rule to find the limit of the sequence $\{a_n\}_{n=1}^\infty$ with $n$-th term $\displaystyle a_n = ...
5
votes
0answers
80 views

$f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
2
votes
1answer
49 views

Metric Spaces: The dist function

Given that $A$ is defined as non-empty subset of $(X,d)$ The distance function is defined as such: $dist(x,A)=$ inf $_{y\in A} \lbrace d(x,y) \rbrace $ Given the above we are asked to prove the ...
0
votes
0answers
43 views

Describe the kernel and the fibers of $\phi$ geometrically (as subsets of the plane).

Define $\phi : \mathbb{C}^{\times} \mapsto \mathbb{R}^{\times}$ by $\phi(a+bi) = a^2 + b^2$. Prove that $\phi$ is a homomorphism and find the image of $\phi$. Describe the kernel and the fibers of ...
0
votes
2answers
29 views

Proof $ GCD(a,b) = GCD(a, b-a) = GCD (a, r_b) $

let $a,b \in \mathbb{N}$ and a < b. let $r_b$ the rest when dividing b through a. (1) If $r_b$ is the rest, then there exists a q so that $ b = q*a + r_b $. (2) Now I show: $gcd(a,b) = gcd(a, ...
3
votes
1answer
47 views

Linear Algebra: Identity map

I was asked to prove that the identity map $id : \Bbb R^n \to \Bbb R^n $ can be represented by the the identity matrix regardless of the basis My Attempt: Let $\mathcal B = \lbrace v_1 , ...,v_n ...
3
votes
0answers
53 views

Proof about sequences of functions.

Is this proof correct? If $\{f_{n}\}$ is a sequence of functions in $C(X,Y)$, $X$ compact, $Y$ complete, and the sequence converges, to $f$, then $K=(\bigcup\{f_n\})\cup \{f\}$ is closed. Proof. ...
1
vote
1answer
25 views

Subgroup generated by d where d is greatest common divisor

Can anyone check if my proof is right? Any help would be appreciated. Problem: Show that the subgroup of $\mathbb{Z}$ generated by any finite set of nonzero integers $n_1,…, n_k$ is $\mathbb{Z}d$, ...
0
votes
0answers
30 views

How to prove the following lemma?

I tried using induction to prove the following lemma Lemma: Let $2R \le D < \alpha R$, for some $\alpha > 2$. If circles are not allowed to overlap, then each circle has less than ...
0
votes
1answer
45 views

Prove that $\phi$ is a homomorphism and find the image of $\phi$. Describe the kernel and the fibers of $\phi$ geometrically.

Define $\phi : \mathbb{C}^x \mapsto \mathbb{R}^x$ by $\phi(a+bi) = a^2 + b^2$. Prove that $\phi$ is a homomorphism and find the image of $\phi$. Describe the kernel and the fibers of $\phi$ ...
2
votes
2answers
55 views

Proving either $x^2$ or $x^3$ is irrational if $x$ is irrational

I had a test today in discrete mathematics and I am dubious whether or not my proof is correct. Suppose $x$ is an irrational number. Prove that either $x^2$ or $x^3$ is irrational. My Answer: ...
1
vote
0answers
44 views

check my answer - Show that $f(A)=trace(A^2)$ is differentiable and find the differential at any point

As topic says, we are given $f: Mat_n(\mathbb R) \to \mathbb R,f(A)=trace(A^2)$ where $A$ is an n by n matrix with real entries. I think I managed to show that $f$ is both differentiable, and find ...
3
votes
1answer
58 views

Describe the fibers of $\phi$ and that $\phi$ is a homomorphism.

Define $\phi: \mathbb{R}^x \mapsto \{\pm1\}$ by letting $\phi(x)$ be $x$ divided by the absolute value of $x$. Describe the fibers of $\phi$ and that $\phi$ is a homomorphism. Need help getting a ...
1
vote
0answers
45 views

Solve an equation with Partial derivatives

If i have the following relation. $$\sin\Theta_{12}\cos\beta=\sin\theta_1\sin\theta_2\sin(\phi_2-\phi_1),$$ and i need to prove that $A_{\Theta_{12}}=0$ considering that $$A_{\Theta_{12}}= ...
0
votes
2answers
78 views

Using Darboux Sums to Prove Upper and Lower Integrals

Define $f:[0, 1]\rightarrow\mathbb{R}$ as \begin{equation} f (x) \equiv \left\{\begin{array}{l l} x & \text{if } x\in [0, 1]\cap \mathbb{Q}\\ 0& \text{if }x\in [0, ...
0
votes
1answer
37 views

Surjective endomorphism of an $R$-module is injective.

I know this is a duplicate question. However, I haven't seen anything that invokes the isomorphism theorem. Here's my idea: By the isomorphism theorem we have that $M/\ker\varphi \cong ...
1
vote
1answer
37 views

Prove $L$ = $\{\langle M \rangle$ | $M$ is a TM over $\{0,1\}$ and $\langle M \rangle \langle M \rangle \notin \mathcal{L}(M)\}$ is undecidable.

Was stuck on this for a bit so I need to know if I am on the right track. To show that $L$ is undecidable we will show that $\overline{L}$ is undecidable instead. Suppose $\overline{L}$ is decidable ...
1
vote
1answer
25 views

Proving $u+\frac1{n}$ is an upper bound

Let $S\subseteq\mathbb{R}$ be non-empty. Show that if $u=\sup S$, then $\forall n\in \mathbb{N}, u-\frac{1}{n}$ is not an upper bound of of $S$ but $u+\frac1{n}$ is an upper bound of $S$. So we ...
2
votes
1answer
34 views

upper bound proof on a nonempty subset of S

Suppose $S\subseteq{\mathbb{R}}$ is nonempty. Show that $u$ is an upper bound of $S$ if and only if $t\in\mathbb{R}, u<t$ imply $t\notin S$ Here's my attempt. $\Rightarrow$ Let $u$ be an ...
1
vote
1answer
54 views

Simple proof involving $\varepsilon$

This seems easy, but now I fell like it's not right. I don't think you can go back along the same line of reasoning when proving an "iff" statement (or can you?)... Show that ...
2
votes
1answer
80 views

Show that $\lim_{x\to c}\frac{f(x)}{g(x)}=\frac{f^{(n)}(c)}{g^{(n)}(c)}$

Suppose $f$ and $g$ are functions such that $f^{(k)}(c)=0$ and $g^{(k)}(c)=0$ for all $k=0,1,\dots,n-1$ $f^{(n)}$ and $g^{(n)}$ exist and are continuous $g^{(n)}(c)\neq0$ where ...
4
votes
2answers
83 views

If $a\lt{b}$ and $c\le{d}$, prove that $a+c\lt b+d$

If $a\lt{b}$ and $c\le{d}$, prove that $a+c\lt b+d$. This seems like a basic proof and I think this is how it goes: $$c \le d, \text{ Given }$$ $$a+c \le a+d$$ $$a+c \lt b+d, \text{ since } a \lt ...
2
votes
0answers
64 views

Prove that $\overline{L}$ is not recognizable by showing that $B_{TM} \le_m L$

$\textbf{Problem}:$ $L$ = $\{\langle M \rangle$ | $M$ is a Turing machine over $\{0, 1\}$ such that for some $x \in \{0,1\}^*$, $M$ does not halt on input $x\}$. $B_{TM}$ = $\{ \langle M \rangle$ | ...
1
vote
1answer
33 views

Find all the $a$ such $539|a3^{253}+5^{44}$

This is what i thought: Given that $539|a3^{253}+5^{44}$ then $11|a3^{253}+5^{44}$ and $7^2|a3^{253}+5^{44}$ using congruences I get: $$a3^{253}+5^{44} \equiv 0 \pmod{7^2}$$ and ...
2
votes
4answers
82 views

$f:X\to Y$, $A,B,\subseteq X$. Show that $f(A\setminus B)=f(A)\setminus f(B)$ iff $f(A\setminus B)\cap f(B) =\emptyset$

I tried to prove this but I am not sure if its correct. Please help me out with any tips or advice on how to improve. Here it is: First let $f(A\setminus B)\cap f(B)=\emptyset$. Now $$f(A\setminus ...
5
votes
1answer
47 views

Show that $f :X \to Y$ is injective iff $f^{-1}f(A))=A$ for all subsets $A$ of $X$ (proof checking)

Show that $f :X \to Y$ is injective iff $f^{-1}f(A))=A$ for all subsets $A$ of $X$. Now I wrote a proof for this theorem and my question is firstly, is it correct? Secondly, since this is my first ...
0
votes
0answers
14 views

Function preserving convergence between metric spaces

Suppose $f: X \rightarrow Y$ is a function between metric spaces $X, Y$ which preserves convergent sequences. Prove that $f$ is continuous. So this is a variation of the typical question that if a ...
1
vote
1answer
40 views

Applications of derivatives: Local maxima, local minima and inflection point

Is that When f(x) is increasing, x is smaller than zero and x is greater than 2? When f(x) is decreasing, x is between 0 and 2 but not equal to 1? The interval where f(x) is concave down is x ...
0
votes
1answer
46 views

Tangent equation divisible by (x-y)

I have attempted this proof but I am not sure is the induction step is correct any assistance would be appreciated also I am not sure if i have proved what I was trying to. Let ...
2
votes
3answers
108 views

Question about the boudary of a set $A \subseteq \mathbb{R}^n $.

let $A \subseteq \mathbb{R}^n$. Let $X = \{ x \in \mathbb{R}^n : \forall \epsilon > 0, \; \; B(x, \epsilon) \cap A \neq \varnothing \; \; and \; \; B(x, \epsilon) \cap ( \mathbb{R}^n \setminus A ) ...
3
votes
1answer
44 views

$\partial A = \overline{A} \cap \overline{ \mathbb{R}^d \setminus A } $

My attempt: ($A \subseteq \mathbb{R}^d$) We know by definition that $\overline{A} = \partial A \cup A $. Hence $\overline{ \mathbb{R}^d \setminus A} = \partial [ \mathbb{R}^d \setminus A ]\cup ...
1
vote
0answers
28 views

Measure Theory - Series of functions

A question from my homework I wasn't sure about: Let $\{f_n\}$ be a sequence of integrable non-negative functions on a measure space $(X,F,\mu)$ s.t. $\int_Xf_n(x)\,d\mu=1$. Is one of the ...
1
vote
2answers
86 views

Using induction, prove that $(-7)^n -9^n$ is divisible by $16$

First of all, I think the problem should be $(-7)^n -9^n$ is divisible by $-16$ because if I test the basis by letting $n=1$, I have $-16$ instead of $16$. Edit: Alright ... I sort of understand why ...
1
vote
2answers
68 views

Correct proof? Linear Algebra

Prove that if $A$ and $B$ are matrices of rank $n$, then $AB$ is of rank $n$. Solution This should be equivalent to proving that the columns $AB$ are linearly independent. $AB = \begin{pmatrix} ...
3
votes
0answers
90 views

Proving a language is not recognizable

I have the following question that I just want to verify I have done correctly. Let $L$, $L_1$, $L_2$ $\subseteq \Sigma^*$ such that $L = L_1 \cup L_2$, and $L_2$ is decidable. Prove that if $L$ is ...
0
votes
1answer
34 views

the boundary of set in euclidean space is closed.

MY Attempt: Let $\partial A$ be the boundary of any set $A \subseteq \mathbb{R}^n$. We show $R^n \setminus \partial A$ is open. Pick $x \in \mathbb{R}^n \subseteq \partial A $. Then by definition, we ...
3
votes
2answers
88 views

Every Cauchy sequence in a metric space $(X,d)$ is bounded.

MY attempt: Suppose $(x_n)$ is a Cauchy sequence in $(X,d)$. Take $\varepsilon = 1 $. Hence, can find $N$ such that $d(x_m,x_n) < 1 $ for all $n,m > N$. Also, we have $d(x_N, x_n) < 1 $ ...
2
votes
2answers
87 views

Are the proofs I made correct?

Edit: Since these are pretty small assignments each and all of the same topic, I've decided to post them into one thread. I hope that's ok. Thank you. Question I have the following assignment: ...