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

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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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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 ...
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1answer
32 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
283 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
65 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 ...
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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} ...
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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
41 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 ...
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1answer
50 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 ...
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176 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: ...
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79 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 ...
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1answer
50 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 ...
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1answer
61 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 $g_{i} : \mathbb{R}^{n} \to \mathbb{R}$ ...
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2answers
66 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 ...
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78 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 ...
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106 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 ...
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2answers
86 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 ...
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1answer
85 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 ...
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1answer
26 views

Improving the proof by contraposition / why it works

This is the problem Prove that if n is an integer and 3n+2 is odd, then n is odd So for this I should take $3n+2$ to be true and assume $\lnot q$, therefore I ...
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2answers
46 views

Summation Simplification

Problem: Let $x$ be a real number. Suppose $x\ne 1$. Let $\in\Bbb N$. Find a simpler expression for the sum $$U=\sum_{k=1}^nk^2x^k\;.$$ For your final answer, put everything over a common ...
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5answers
62 views

Proof that the $\lim_{x \to 0}f(x)$ does not exist.

Prove that the $\lim_{x \to 0}f(x)$ does not exist. $$ f(x) = \begin{cases} \space\space\space 1 & \text{if } x \text{ is rational}\\ -1 & \text{if } x \text{ is irrrational} ...
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86 views

Show by induction on $n$ that:

$$1^4 + 2^4 +\cdots+n^4=\frac{n^5}{5}+\frac{n^4}{2}+\frac{n^3}{3}-\frac{n}{30}$$ I proved true for case $n=1$ , assumed true for $n=k$ , but cannot get things to work out. I tried putting the right ...
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3answers
266 views

Formal logic behind defining variables in a proof (not E.I. or U.G, etc.)

This is something I have been curious about and hopefully has a simple answer. Often when looking at proofs I will come upon a step that goes along the lines of "define y = ..." and then proceed to ...
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1answer
91 views

Recursive Definition of Is Equal To

I'm working through some of the intro problems in Sudkamp's Languages and Machines (basically an intro book to finite automata, context free grammars, Turing machines, etc), and I'm struggling a bit ...
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3answers
901 views

Let A and B be sets. Prove that A = B iff the power set of A is equal to the power set of B.

I am an undergraduate student. Please tell me if my proof is correct. Thanks! Let A and B be sets. Prove that A = B iff the power set of A is equal to the power set of B. Assume that A and B are ...
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178 views

For a, b ∈ N, let A and B be the sets of all integer multiples of a and b. Prove that for all a,b ∈ N, a = b iff A = B.

I am an undergraduate student. Please tell me if my proof is correct. Thanks! For a, b ∈ N, let A and B be the sets of all integer multiples of a and b. Prove that for all a,b ∈ N, a = b iff A = B. ...
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1answer
865 views

Prove that if x ∉ B and A ⊆ B, then x ∉ A

I am an undergraduate student and I am wondering if the strategy and the writing of this proof are correct. Please help me! Prove that if x ∉ B and A ⊆ B, then x ∉ A. Assume that x ∉ B and A ⊆ B, ...
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1answer
25 views

Proving convergence theorems

Suppose that ${a_n}$ converges to a. prove the sequence ${ca_n}$ converges to ca, where c is any constant. Here is the start of my proof: By definition of convergence of a sequence, $|ca_n - ca| ...
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110 views

Computation of the probability of Bernoulli trials

I want to solve the following : Independent trials that result in a success with probability $p$ and a failure with probability $1-p$ are called Bernoulli trials. Let $P_n$ denote the probability ...
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1answer
84 views

Generalising mean value theorem for integrals

I want to generalize mean value theorem for Riemann integrals to $\Bbb R^n$, but I do not know how to formulate it. Can someone please help me with formulating the theorem? I think I can prove it ...
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95 views

Prove that each diagonal of a quadrilateral lies either entirely in its interior or entirely in its exterior.

From Kiselev's Planimetry problem 55. Here's my attempt at a proof, I'd appreciate feedback on how to get better beyond whether or not it's correct, as I have very little experience and am still ...
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1answer
50 views

Partitioning in groups - Combinatorial Proof

Define $G(n;k)$ as the number of possibilities to portion out $n$ distinguishable elements into $k$ groups. For each group we count the number of unique orders. Two orders are different if they ...
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208 views

Proof that the $\lim\limits_{x \to 2}\dfrac{1}{x} = \dfrac{1}{2}$ using the $\epsilon$-$\delta$ definition of limits (verification).

Prove that the $\lim\limits_{x \to 2}\dfrac{1}{x} = \dfrac{1}{2}$ using the $\epsilon-\delta$ definition of limits. $$ \\ \begin{align} \\ &\textrm{Let } \forall \epsilon > 0 \\ ...
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126 views

Can we prove that there is no limit at $x=0$ for $f(x)=1/x$ using epsilon-delta definition?

The $\varepsilon-\delta$ definition of limit is: $$\lim_{x \to a}f(x)=l \iff (\forall \varepsilon>0)(\exists \delta>0)(\forall x \in A)(0<|x-a|<\delta \implies |f(x)-l|<\varepsilon)$$ ...
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1answer
41 views

Show that if $x^2 y=2x+y$, then if $y \neq 0$ then $x \neq 0$

Prove if $x^2 y=2x+y$, then if $y \neq 0$ then $x \neq 0$. Obviously, $x,y \in \mathbb R$. I know this is rather simple. It is more about the process than this example. Is it logically correct to do ...
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1answer
314 views

Showing that the intersection of these two subspaces is {0}

This is something that came up while I was working on a problem. I approached it a little bit differently, and I wanted to see if this approach could work. We suppose that we have two vector spaces ...
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5answers
57 views

Two Questions: (1) Is My Proof Method Legitimate, and (2) How Might Proof by Cases be Accomplished?

Prove 5x-4 is even iff 3x+1 is odd. I have two questions: First, for example, could we assume that 5x-4 is even such that 5x-4 = 2k, for some integer k and then manipulate the above equation to ...
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1answer
75 views

When should I use “let” “put” “be” in a proof?

From reading some instructions I could find online, I could understand that this isn't universally agreed upon, but in a course I'm taking now the professor insists on a particular connection between ...
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168 views

Proof : Do 4 days fall on the same day?

I was working my way through some discrete math proof examples from Discrete Math by Rosen and being a newbie am stuck on this problem : Show that at least four of any 22 days must fall on the ...
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2answers
542 views

Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x+y=z

This is a homework assignment, please tell me if my proof is correct! Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x + y = ...
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36 views

How a proof like this should be written?

I'm studying with Apostol's Calculus I, and it has required me to carry out some proofs, which I'd never done before (which has turned out to be pretty neat, by the way). When proving a statement ...
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1answer
34 views

The dimension of a set of points

This is part of a step for a larger problem that I am working on, but I just wanted to make sure that I said this part correctly. I have the set of points $\{(x,x-y,y,z,0): x,y,z \in \mathcal{R}\}$ ...
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1answer
96 views

How do I write this proof? [closed]

I've tried everything, but I'm just not getting it. The steps I tried were just blatantly wrong. Let $T:\mathbb{R}^m \rightarrow \mathbb{R}^n$ be a linear transformation. Prove that $T$ is ...
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1answer
89 views

Prove that $(z^3-z)(z+2)$ is divisible by $12$ for all integers $z$

I am a student and this question is part of my homework. May you tell me if my proof is correct? Thanks for your help! Prove that $(z^3-z)(z+2)$ is divisible by $12$ for all integers $z$. ...
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1answer
83 views

Using contrapositive to Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7 [duplicate]

so I know that the contrapositive will be something like; If all the numbers are greater than or equal to 7, then the average cannot be less than 7. How do i go about proving it from there? or is ...
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1answer
75 views

How to formal prove |set 1| is less than or equal to |set 2| + |set 3|

I could prove |set 1| $\le$ |set 2|by defining a function $f$ such that $f$ is a one-to-one function Suppose X = {1, 2, 3} and Y = {D, B, C, A} or X = {1, 2, 3, 4} and Y = {D, B, C, A} and ...
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2answers
309 views

Proof by induction (combinations)

We are supposed to prove this via induction. I originally solved it with simple algebra, showing that $n = n$ and $n+1 = n+1$, but a friend told me that wasn't really solving it by induction and said ...
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2answers
607 views

Proof by induction and combinations

I think I am stuck on this, I am not sure if I'm going down the correct path or not. I am trying to algebraically manipulate $p(k+1)$ so I can use $p(k)$ but I am unable to do so, so I am not sure if ...
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1answer
23 views

Example of a set with some caracteristics.

My question this time is, I am asked to find an example of a subset of $\mathbb{R}^{n}$ such that it is closed but not bounded, and all continous function defined there are also uniform continous, but ...
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89 views

Is $\frac{1}{z}$ analytic?

Suppose I am to prove: $$f(z) = \frac{1}{z}$$ is analytic everywhere. I see there is an obvious discontunuity at $z=0$, but we can apply the residue theory, which means f(z) is indeed analytic $$ ...