For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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10 views

If $a, b, … e$ $\in \mathbb{R}$ and $a \neq 0$, if $ax + by = c$ has the same solution set of $ax + dy = e$, then these equations are the same

I have an exercise in my last assignment for liner algebra class, where I have to prove that 2 equations are the same. The problem is the following: Prove that, if $a, b, ... e$ are real ...
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2answers
25 views

Topology proof: dense sets and no trivial intersection

I was wondering if this proof of this basic topological result concerning the closure works. Proposition: Let $A \subseteq (X,\tau)$. Then, $A$ is dense in $X$ if and only if every non-empty open ...
2
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3answers
113 views

Proof of the Product Limit Law

Theorem: $$\lim_{x \to a} f(x) = L$$ $$\lim_{x \to a} g(x) = M$$ Then: $$\lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = LM$$ Obviously, $$|f(x) - L| < \epsilon$$ $$|g(x) - M| < ...
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1answer
40 views

Irrationality proof trick with Mod [duplicate]

You will see here: Bill Dubuque's Slick $\sqrt{3}$ irrationality proof What is the trick with modulus for proving irrationality? What about $\sqrt{2}$ Can you prove this is irrational by that ...
5
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1answer
55 views

need help proving an interval

I am trying to proof $$\frac {1} {ek} \le \frac {1}{k} (1 - \frac {1}{k} )^{k-1} \le \frac {1}{2k} $$ for k>=2 to prove this I first multiply by k getting $$\frac {1} {e} \le \left(1 - \frac ...
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2answers
24 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 ...
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0answers
13 views

Can this proof-by-cases of 10-tuple patterns be generalized to all $k$-tuples?

Lemma: All Palindrimic, Prime Patterns Sieve's $10$-tuple patterns contain at least one $1$. Proof: The P$\mathbb{P}$P$^{[2]}$ Sieve produces $11$ unique, $10$-tuple patterns. We use the ...
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1answer
26 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
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1answer
22 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 ...
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0answers
42 views

squeeze theorem - math

I am trying to prove the following: 1/ek <= (1/k)(1-(1/k))^(k-1) <= 1/2k for k>=2 in doing so I tried induction proof, and contradiction and it didn't work, it gets too complicated... Then ...
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0answers
22 views

Addressing a conjecture with different “strengths”

I plan on writing a conjecture in a publication I am developing. However, the conjecture has different "strengths" to it, take this example: Conjecture 1.1. $\exists a$, such that $a$ is a solution ...
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2answers
19 views

Proof regarding continuous random variables? Show $P(\{a<X\leq b\}) = F_X(b) - F_X(a)$ [on hold]

I understand that an integral function is used here but im pretty hazy so a breakdown of this question would be good Show that: $P(\{a<X\leqslant b\}) = F_X(b) - F_X(a)$
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1answer
39 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 ...
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5answers
100 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 ...
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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) = ...
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0answers
22 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 ...
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2answers
77 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 ...
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1answer
23 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 ...
5
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2answers
81 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
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1answer
41 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: ...
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3answers
21 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 ...
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1answer
36 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 ...
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1answer
29 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 ...
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1answer
28 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
22 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 ...
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1answer
42 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 ...
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1answer
23 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 ...
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2answers
38 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
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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 ...
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3answers
35 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 = ...
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2answers
243 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 ...
1
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1answer
28 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 ...
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1answer
37 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
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1answer
51 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 ...
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1answer
39 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.
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4answers
35 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?
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1answer
33 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 ...
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1answer
49 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 ...
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1answer
30 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
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1answer
19 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: ...
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2answers
36 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 ...
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0answers
122 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 ...
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3answers
63 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
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1answer
24 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 ...
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2answers
54 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?
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1answer
47 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
13 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} ...
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votes
1answer
27 views

Proof that point (m,n) is on parabola [closed]

If a, b are positive rational numbers and c is an integer, then there is a pair of integers, $p$ and $q$, such that the point, $(m, n)$ , is on the graph of the parabola $$y=ax^2 + bx + c$$. Hint: ...
1
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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."?
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1answer
34 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 ...