For questions about the formulation of a proof, not about the mathematics behind it.

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0
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2answers
113 views

order of subgroup same as order of group(finite groups)

If I have order of a subgroup C of same order as group G I want to prove that G = C. One inclusion is obvious C $\subset$ G the other inclusion we can get by a bijection f : G $\rightarrow$ C hence ...
1
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1answer
47 views

Proof that the discrete metric $d$ is complete in $\mathbb{N}$

This is an attempt of a proof of a rather basic result. Proposition: The discrete metric $d$ is complete in $\mathbb{N}$. Proof: Let $x_n$ be an arbitrary sequence in $\mathbb{N}$ endowed ...
2
votes
1answer
38 views

why conventional approximation method is true?

why the text book method for finding the fitting curve is right ? we have n data we want to approximate with a polynomial of degree m $P_m(x)$ (m < n-1). and of course $E = \sum_{i=1}^m ...
0
votes
1answer
52 views

Algebraic Properties of the Integral

Prove that $$\frac{1}{3\sqrt{2}} \leq \int_0^1 \frac{x^2}{\sqrt{1+x^2}}\space dx \space \leq \frac{1}{3}$$ Use: If $f_1(x)$ and $f_2(x)$ are integrable on $[a, b]$ and $f_1\leq f_2$ then $$\int_a^b ...
2
votes
5answers
114 views

Show that $f''(x) = 0$ for some $x$

Let $f$ be a twice differentiable function with the following properties: $f(x) > 0$ for $x \ge 0$. and $f$ is decreasing, and $f'(0) = 0$. Prove that $f''(x) = 0$ for some $x > 0$. The ...
0
votes
1answer
13 views

Proof of a polynomial given parameters

Let $a_1, a_2, ... a_n$ and $b_1, b_2, ... b_n$ be given numbers. If $x_1, x_2, ... x_n$ are distinct numbers, prove that there is a polynomial function $f$ of degree $2n - 1$, such that $f(x_j) = ...
0
votes
0answers
53 views

Corollary of the inverse function theorem

Let $U\subset \mathbb{R}^{n}$ and $ f:U\to \mathbb{R}^{n}$ injective and class $C^{1}$ such that $\det f'(x)\not=0$ for all $x \in U$. Show that $f(U)$ is open and $f^{-1}:f(U)\to U$ is ...
1
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2answers
100 views

when does one use the word 'fact' in mathematics

I am having trouble using the word 'fact' when speaking/writing Mathematics. For instance, suppose we have proved a new theorem, I then apply the theorem to deduce some other results. Can I regard ...
0
votes
1answer
27 views

Product of a prime and other number expressed in two ways are equal

Let $p \in \mathbb P$ be an odd prime and let $1\leq a \leq p-1$ be such a number that $$a p = \left\lceil \sqrt{a p}\right\rceil ^2-\left\lceil \sqrt{\left\lceil \sqrt{a p}\right\rceil ^2-a ...
8
votes
3answers
183 views

Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.

Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$. My work: So I think I have to do a proof by induction and I just wanted some help editing my proof. My ...
3
votes
1answer
90 views

real analysis converging proof using Abel's formula.

Suppose that $\Sigma_{k=1}^\infty a_k$ converges. Prove that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$. Attemtp: ...
1
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3answers
174 views

How to prove that the inverse image of the image of a set is a subset of the set.

What I need to prove is the following: Let $f:X\to Y$ and $A\subset X$. Prove that $f^{-1}(f(A))=A$ for all $A$ if and only if $f$ is injective. So, I realize that I have to prove both directions ...
0
votes
1answer
41 views

Prove a consequence of the multivariable version of the inverse function theorem

The exercise is the following: Let $f:\mathbb{R}^{n} \to \mathbb{R}^{n}$ that is class $C^{1}$ such that there exists $c >0$ such that $$|f(x) - f(y)| \ge c|x-y|$$ for all $x,y \in ...
0
votes
1answer
33 views

Use $\epsilon$ - $\delta$ definiton to prove $\lim_{z\to 1}$ $\frac{z+2}{z+3i}$ $=$ $\frac{3}{1+3i}$

So i need need to prove that $$\lim_{z\to 1} \frac{z+2}{z+3i} = \frac{3}{1+3i}$$ So far my understanding is that we want to calculate $\left|{f(z)-z}\right|$ and manipulate it in such a way that we ...
-1
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1answer
102 views

How to prove the inductive step in this Mathematical induction problem?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 6, pg 342]. Problem: a) Determine which amounts of postage can be formed using just $3$-cent and $10$-cent ...
2
votes
2answers
62 views

How to come up with relation in induction hypothesis for strong induction

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 2, page 341]. Problem: Let $P(n)$ be the statement that a postage of n cents can ...
1
vote
1answer
49 views

Attempting a discrete proof: Not sure what I am doing wrong?

So this is an exercise that is a supplement to my studies in discrete math, I want to understand what my error is. The online training drill I am using reports the below is incorrect / or as we would ...
0
votes
1answer
256 views

How to show the inductive step of the strong induction?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 2, pg 341]. Problem: Use strong induction to show that all dominoes fall in an infinite arrangement of dominoes if ...
0
votes
2answers
57 views

How to get $k^{k + 1} + k^k$ to equate $(k+1)^{k+1}$?

This is a problem from Discrete Mathematics and its Applications Let $P(n)$ be the statement that $n!<n^n$, where $n$ is an integer greater than $1$. $\quad(a)$ What is the ...
0
votes
1answer
19 views

Connectedness of an Image

Given the sets $S_1 = \{ z \in \mathbb{C}; Im(z) > 1 \}$, $S_2 = \{z \in \mathbb{C}; Im(z) < -2\}$, is the image of $S_1 \bigcup S_2$ connected (path connected) under the map $w = z^2$? My ...
2
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3answers
42 views

How to show no other elements besides $\pm 1$ will be in the kernel of $h: \mathbb Z_p^* \rightarrow \mathbb Z_p^*$; $h(\bar{a}) = \bar{a}^2$.

Let $p$ be a prime and let $h: \mathbb Z_p^* \rightarrow \mathbb Z_p^*$ be defined by $h(\overline{a}) = \overline{a}^2$. Since $h(\overline{xy}) = \overline{xy}^2 = \overline{x}^2 \overline{y}^2 = ...
0
votes
2answers
318 views

Putnam 1990 A1 Induction Help

A1. $(150,9,1,0,0,0,0,0,1,1,6,33)$ Let $$T_0=2,\quad T_1=3,\quad T_2=6,$$ and for $n\ge3$, $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.$$ The first few terms are ...
2
votes
1answer
122 views

The order of a $k$-cycle in $S_n$ is $k$.

Here's what I have right now: The order of a $k$-cycle in $S_n$ is $k$. Proof. Let $\sigma$ represent the $k$-cycle $$\sigma=(x_1 \ x_2 \ \cdots \ x_k)$$with distinct elements $x_i$. Note that the ...
0
votes
1answer
60 views

Will these rules hold for multi-sets (bags)?

I have proved that RHS = LHS, but I don't know whether that is what is being asked, or htey want something else. For example, for No. 2, I have proved the relationship like; $$ \begin{split} (R\cup ...
1
vote
1answer
72 views

Linear Mapping Proof with Kernel & Range

Given that $L$ is a linear map such that $L:\Bbb R^n\to \Bbb R^m$, and $\ker(L) = \{\mathbf{0}\}$ and $\operatorname{range}(L) = \Bbb R^m$, show that $m = n$. So far, I've tried to use the ...
0
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1answer
85 views

Proving set properties?

I am stuck with proving that RHS = LHS. I don't know where to begin and how prove the questions below.
0
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4answers
42 views

Convergence of $\sum_{n=1}^\infty\frac{3}{2n^{p+1}}$

Convergence of $\sum_{n=1}^\infty\frac{3}{2n^{p+1}}$ . What will be the best proof for convergence of this series, which criterion will be the best?
0
votes
1answer
40 views

Proof by contradiction, fields

Given the field $\mathbb{K}:=\{a+b\sqrt{2}: a,b\in \mathbb{Q}\}$, how would I prove that every $x\in \mathbb{K}$ is uniquely representable in this way: $x=a+b\sqrt{2}$, with $a,b\in \mathbb{Q}$? I ...
0
votes
1answer
55 views

Show that $A \in GL_2(\mathbb Z_2)$ if and only if $\det(A) \neq 0$ and $A^{-1} = \det(A)^{-1} B$.

Where $B = \begin{bmatrix} [d] & [-b] \\ [-c] & [a] \end{bmatrix}$ The way I solved this problem is That I first got all the elements of $M_2(\mathbb Z_2)$ and then considered the ...
0
votes
1answer
36 views

Lemma concerning compatibility of words (formed by a term algebra)

I need to prove the next lemma regarding compatibility of words in term algebras, that includes 3 parts: $u,v$ are compatible iff $u^ \smallfrown w_1= v^ \smallfrown w_2$. If $u_1u_2$ and $v_1v_2$ ...
1
vote
1answer
24 views

Which statement should I prove?

I have an exercise, but my problem is not directly related to how to solve it, but which statement do I have to prove. The following is the exercise: b) Prove the following: ...
0
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2answers
44 views

How would I say that two elements do not belong to the same set?

Let's say I have two lists, X, Y. X = {limegreen, forestgreen, seagreen}, Y = {babyblue, navyblue, ultramarineblue} And I have the elements $d_0, d_1, d_2, ..., d_n$. I want it so no two ...
1
vote
0answers
148 views

Linear Algebra Proof for matrices

Could someone possibly help me in proving this: Let $A$ be the augmented $m \times (n + 1)$ matrix of a system of m linear equations with $n$ unknowns. Let $B$ be the $m \times n$ matrix obtained ...
1
vote
3answers
102 views

Geometric proof: Legs intersect on CM (median of triangle)

$M$ is the midpoint of $AB$ in the triangle $\triangle ABC$. The angle $\angle ACM$ is copied and drawn on the leg $AB$ in $A$. The angle $\angle MCB$ is copied and drawn on the leg $BA$ in $B$. The ...
3
votes
1answer
31 views

Show that $\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}$ where $|x| < 1$ is not uniformly convergent

Show that $\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}$ where $|x| < 1$ is not uniformly convergent My professor's proof is as follows: So we know that the radius of convergence is $R = 1$. Now ...
3
votes
2answers
225 views

Understanding how to prove limit theorems for sequences.

How do you know what arbitrary value to choose for epsilon as in this document, and when should the triangle inequality be considered when writing your proof?
1
vote
1answer
60 views

Uniform convergence of $\sum_{k=1}^{\infty}\frac{2^{k-1} x^{2^{k-1}-1}}{1+x^{2^{k-1}}}$

Does $\sum_{k=1}^{\infty}\frac{2^{k-1} x^{2^{k-1}-1}}{1+x^{2^{k-1}}}$ converges uniformly. $-1<x<1$ I have tried to bound it by Weierstrass M-Test but haven't been successful. I have also tried ...
1
vote
0answers
14 views

Finding specific functions $g_i$ in $f(x)= \sum_{i=1}^{n} x^{i}g_i $

Let $f: \mathbb{R}^{n} \to \mathbb{R}$ be differentiable (may be not in $C^{1}$) and $f(0)=0$, show that there exists $g_{i} : \mathbb{R}^{n} \to \mathbb{R}$ such that: $$f(x)= \sum_{i=1}^{n} ...
1
vote
2answers
31 views

Is it fine to say isomorphic or should one say isomorphic to each other?

Is it fine to say "Groups $A$ and $B$ are isomorphic." or should one say "Groups $A$ and $B$ are isomorphic to each other."?
1
vote
1answer
40 views

Induction help with final answer

Use induction to prove that for any complex number $z$ that does not equal $1$ and integer n is greater or equal to 1: $$ 1+z+z^2+...+z^n = \frac{1-z^{n+1}}{1-z} $$ So far for the base case I used ...
1
vote
1answer
48 views

Estimating distance between two functions

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. I am trying to prove that $F$ is closed in $BC(\Bbb R, \Bbb R)$, where $BC$ is the space of ...
4
votes
0answers
174 views

$\zeta(2)=\frac{\pi^2}{6}$ proof improvement.

Recently in one of my calculus exercise I have made out a (quite novel to me) proof for $\zeta(2)=\frac{\pi^2}{6}$ via the famous infinite product below: ...
5
votes
0answers
72 views

Proving not equicontinuity in $\Bbb R$ but equicontinuity in any other closed subset of $\Bbb R$

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. Prove that $F$ is not equicontinuous on $\Bbb R$ but equicontinuous on $[−a, a]$ for any $a ...
0
votes
1answer
47 views

Showing a Set is Dense in $C[a,b]$

Show that $C_0[a,b]:= \{f \in C[a,b] : f(a)=f(b)=0 \}$ is $\| \cdot \|_2$-dense in $C[a,b]$. Is it also $\| \cdot \|_{\infty}$-dense in $C[a,b]$? In the relevant chapter of my textbook, I am given ...
1
vote
1answer
59 views

A decomposition of a differentiable function

this time I want to solve this problem: Let $f: \mathbb{R}^{n} \to \mathbb{R}$ be differentiable (may be not in $C^{1}$) and $f(0)=0$, show that there exists $g_{i} : \mathbb{R}^{n} \to \mathbb{R}$ ...
2
votes
2answers
64 views

Proof by Contradiction Minimum Value Proof $f(x)$

Focusing on $x=a$ first. My Proof: Assume $f'(a) < 0$ $f(x) \le f(x_1)$ for all $x$, this follows from the extreme value theorem. $$f'(x_1) = 0$$ Because it is a maximum. $$\exists x_4 ...
3
votes
0answers
76 views

Help with a trigonometric proof, please?

Hexagon $ABCDEF$ is inscribed in the circle of radius $R$ . $AB=CD=EF=R$. Points $I$, $J$, $K$ are the midpoints of segments $\overline{BC}$, $\overline{DE}$, $\overline{FA}$ respectively. Then ...
1
vote
3answers
94 views

If f and g are continuous on {x | a <= x <= b} and the integral of their product is 0, prove that f(x) = 0

Suppose f is continuous on $I=\{x|a \leq x \leq b\}$ and $\int_a^b \! f(x)g(x) \, \mathrm{d}x = 0$ for all functions $g$ which are continuous on $I$. Prove that $f(x) = 0$. In this case, the integral ...
2
votes
2answers
80 views

Prove an equivalence involving $ (\cup \mathcal{F})\cap(\cup \mathcal{G}) \subseteq \cup (\mathcal{F}\cap\mathcal{G}) $.

This question is two-fold. First, I'm looking for feedback on a proof I wrote for the following problem. It's from exercise 18 in section 3.4 of Velleman's How To Prove It. The section deals with ...
0
votes
1answer
68 views

Question regarding normal spanning trees and a proof of existence

I'm reading about normal spanning trees in the Diestel book and i am somewhat confused by a number of things i'll try and work in chronological order. The first thing you need to know is about a tree ...