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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0
votes
3answers
55 views

Proving the sandwich theorem for $\lim_{n \to \infty} c_n$ if $a_n \leq c_n \leq b_n$ and $a_n, b_n \to c$

Suppose $\lim\limits_{n \rightarrow \infty} a_n =\lim\limits_{n \rightarrow \infty} b_n = c$ and $a_n \le c_n \le b_n$ for all $n$. Prove that $\lim\limits_{n \rightarrow \infty} c_n = c$. How ...
0
votes
4answers
63 views

Proof roots of unity being in $\mathbb R$

Let $n \in \mathbb N$ even, and be $w,z \in \mathbb G_n$ primitives. Proof that $(w+z)^{n/2} \in \mathbb R$. Ok, as I didn't really know how to start, I tried several things, such using the Binomial ...
2
votes
1answer
49 views

Does all non-monotonic continues functions have $x_0 \in \mathbb{R}$ such that $f'(x_0)=0$?

Given $f\colon\mathbb{R} \to \mathbb{R}$, $f$ is differentiable on $\mathbb{R}$ and the $\lim_{x \to \infty}f(x)$ does not exists . show/prove formally that there exists $x_0 \in \mathbb{R}$ such ...
1
vote
1answer
54 views

Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$

$f(x)$ is continuous on $[a,b]$. Now we define a new function $M(t)$, for every $t\in[a,b]$ $$M(t) = \sup_{a \leq x \leq t} f(x).$$ Prove formally that $M(t)$ is continuous on $[a,b]$. (sup = ...
0
votes
1answer
89 views

Prove this limit

If $\lim_{x\to a}f(x)=L>0.$ Prove $\lim_{x\to a}\sqrt(f(x))=\sqrt(L)$. I know that we have: |$\sqrt(f(x)-\sqrt(L)|=|(f(x)-L)/\sqrt(f(x)+\sqrt(L)|\le|(f(x)-L)/L|<|(f(x)-L)|<\epsilon$. ...
1
vote
2answers
164 views

Linear surjective isometry then unitary

Basically what I'm trying to show is $\forall h_1, \ h_2 \in \mathscr{H}$ and $U: \mathscr{H} \rightarrow \mathscr{K}$ then $\langle Uh_1, \ Uh_2\rangle_\mathscr{K} = \langle h_1, \ h_2 ...
0
votes
5answers
437 views

Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$

Prove that $$\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$$ The exercise provides the following hint: $\,\,\displaystyle{n \choose k}={n\choose n-k}$. Any help?
0
votes
2answers
139 views

How to prove the equation |xy|=|x||y| if we assume x and y are real numbers by using analysis. [closed]

Prove that if x and y are real numbers, then |xy|=|x||y|. Hint check all the cases. I tried assuming the left hand side equals the right hand side if we remove absolute values. Also, tried using the ...
1
vote
1answer
27 views

Need help with a proof

Let $m, n \in \mathbb{N}$. If $n$ is divisible by $m$, then $m \le n$. So far I have: Let $m,n \in \mathbb{N}$ and assume that $n$ is divisible by $m$. Therefore, there exists $j \in \mathbb{Z}$ ...
0
votes
1answer
87 views

Writing my first mathematical paper

I'm an undergrad student at a community college, and I am currently taking differential equations. My professor is offering people to hand in math papers for either extra credit or even complete test ...
3
votes
3answers
160 views

Proof of something that doesn't exist

Let $\lfloor x \rfloor$ be the greatest integer function. Show that the $\lim_{x\to 2} \frac{1}{\lfloor x \rfloor}$ does not exist. So far I have: Assume the limit exists. Choose $\epsilon ...
0
votes
1answer
57 views

How to show that if $ p \equiv 1,3 \pmod 8$ then there exists a $u,v \in \mathbb Z: u^2 + 2v^2 = p$

I'm trying to show this statement: $$p \equiv 1,3 \pmod 8, \; \; \exists \; u,v \in \mathbb Z : u^2 + 2 v^2 = p.$$ I believed I proved it the other direction using the ring $\mathbb Z{\sqrt{-2}}$. ...
3
votes
2answers
72 views

Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function. Prove formally that $P$ is onto $\mathbb{R}$

Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$ my strategy so far ....... polynomial function is ...
4
votes
2answers
115 views

formal proof from calulus

Given $f:R \to R$, $f$ is differentiable on $R$ and $\lim_{x \to \infty}(f(x)-f(-x))=0$. I need to show that there is $x_0 \in R$ such that $f'(x_0)=0$ I am trying to prove it by contradiction .... ...
0
votes
0answers
46 views

How to write down this proof about a graded ideal in multilinear algebra?

I have a very simple question, but since this is the first time I'm dealing with graded ideals and so on it seems more difficult than it really is. Suppose $V$ is a finite dimensional vector space ...
2
votes
2answers
68 views

My problem in understanding the minimal counterexample technique

If minimal counterexample method of proof is to assume to opposite of an argument is true and then finding a counterexample for the opposite and then concluding the validity of the original argument, ...
-2
votes
2answers
27 views

Proof Involving sum

Can someone point me in the right direction, should be able to figure it out. Let $x\in\mathbb{R}^+$. Prove that $x + \frac{1}{4x} \geq 1$ Thanks
-3
votes
2answers
194 views

Proof Involving Rational Numbers [duplicate]

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this ...
1
vote
1answer
62 views

Linear Independent proof

In my Linear Algebra class we define Linear dependence as follows: If $F$ is a field and $V$ is a vector space over the field $F$. The set $A = {\lbrace v_1,v_2,...,v_k \rbrace}$ where ...
-2
votes
1answer
777 views

Rational Number Proof [duplicate]

Stuck on a tutorial question trying to study for a test. The question is : Consider the following statement: "Between any two different rational numbers, there are at least two different rational ...
1
vote
3answers
122 views

How is the Logarithm derived from the exponential function? (aren't they inverses?)

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series ...
0
votes
1answer
38 views

Need help proving the following:

Any help at all would be great. Thank you very much. For all $m,n,p \in \mathbb{Z}$, If $p<0$ and $mp<np$ then $n<m$
20
votes
7answers
2k views

LaTeX/TeX Vs. Mathematica for Typesetting

I know Mathematica like the back of my hand, but I do not know a speck of $\LaTeX$ or $\TeX$. With regards to mathematical typesetting, is there something significant I can do in $\LaTeX$/$\TeX$ that ...
0
votes
3answers
774 views

Sum of $k {n \choose k}$ is $n2^{n-1}$

Proof that $\suṃ̣_{k=1}^{n}k {n \choose k}$ for $n \in \mathbb N$ is equal to $n2^{n-1}$. As a hint I got that $k {n \choose k} = n {n-1\choose k-1} $. I tried solving this by induction but, in the ...
0
votes
1answer
63 views

Group Order and Least Common Multiple

Let $G_1,G_2,...G_n$ be groups. Show that the order of an elements $(a_1,a_2,...a_n)$ $\in$ $G_1 \times G_2 \times ... \times G_n$ is lcm($o(a_1),...,o(a_n))$ I know I need to use the fact that the ...
1
vote
1answer
29 views

Multiplication cannot be obtained from zero, successor, and identity by composition without recursion

The task is to show that multiplication cannot be obtained by zero, successor, or identity functions by composition without using recursion at least twice. I'm primarily confused because it doesn't ...
1
vote
1answer
47 views

I don't understand part of a proof

I was reading a proof in my textbook today and couldn't figure out why this is true: $$ nq - mp = nq -mq +mq - mp$$ Any help would be appreciated.
0
votes
2answers
48 views

Need help with a math proof

Any help would be greatly appreciated. Let $m,n,p,q \in \mathbb{Z}$. If $0 < m < n$ and $0 < p < q$ then $mp < nq$.
1
vote
1answer
38 views

Proving Direct Sum

Claim. Let $V$ be a vector space over $F$, and suppose that $W_1$, $W_2$, and $W_3$ are subspaces of $V$ such that $W_1 + W_3 = W_2 + W_3$. Then $W_1 = W_2$. I know that this claim is false, but ...
0
votes
2answers
44 views

Proving convergence to a certain limit

Suppose that the sequence $(X_n)$ has the following property: there is a real number $a$ such that there are infinitely many $n$ for which $X_n = a$. Prove that, if $X_n$ converges at all, its limit ...
3
votes
2answers
64 views

Prove that the function $f:G \rightarrow G$ given by $f(a^i)=b^i$ is an automorphism of $G$

Assume that $a$ and $b$ are both generators of the cyclic group $G$, so that $G=<a>$ and $G=<b>$. Prove that the function $f:G \rightarrow G$ given by $f(a^i)=b^i$ is an automorphism of ...
1
vote
1answer
36 views

Injection proof

Prove that for any $A, B \subseteq X$ we have $f(A \cap B) = f(A) \cap f(B)$, then $f$ is an injection. I get stuck at the step where $f(w) = y = f(z)$, since I am trying to prove it is injective, I ...
1
vote
1answer
93 views

Tangent Cone is a cone?

I first give two definitions. Def1: A set $S$ is a cone if $x \in S, \lambda \geq 0 \implies \lambda x \in S$. Def2: Let $S$ be any set (we may assume $\mathbb{R}^n$ with the usual Euclidean ...
0
votes
1answer
16 views

Ideals of set of functions from real to real

I'm looking to prove the following is an ideal of the set of functions from real numbers to real: a)the set of all f such that f(x) = 0 for every rational x b) the set of all f such that f(0) = 0
2
votes
1answer
26 views

Find m $\in \mathbb N$ that as condition is product of $3$ primes and the equation $x^2 +1 \equiv 0 \pmod{m}$ has $8$ solutions modulo $m$

What I thought was that if $$x^2 +1 \equiv 0 \pmod{m}$$ has to be met, then $$x^4 \equiv 1 \pmod{m}$$ it's also a condition. Then I looked for primes that hold to this condition, and using brute ...
7
votes
8answers
271 views

Proof of Divisibility of $n(n^2+20)$ by 48.

This is a question from Bangladesh National Math Olympiad 2013 - Junior Category that still haunts me a lot. I want to find an answer to this question. Please prove this. If $n$ is an even ...
0
votes
1answer
86 views

$A=(A \cap B) \cup(A \cap B^\mathsf{c}) $

I would like to know if this proof is correct. If not, what would I have to change to make it rigorous? This set equality seems really obvious, and because of that I am not sure if I have given enough ...
0
votes
0answers
110 views

explain this histogram

A medical researcher measured systolic blood pressure in $100$ middle aged men. the results are displayed in the accompanying histogram; note that the distribution is rather skewed. According to the ...
3
votes
0answers
143 views

Substitute Proof by Contradiction/Negation with Direct Proof or Proof by Contrapositive

When do proof by contradiction, or by negation, or direct proof, or proof by contrapositive all work equally well? When can any one be chosen as the proof form? I'm interested in a pragmatic answer ...
1
vote
5answers
119 views

Showing something is not onto?

Quick question..: If I have a linear transformation $T:\mathbb R^2 \to \mathbb R^4$, can I say that since $\mathbb R^4$ is greater than $\mathbb R^2$, it can never be onto - since the elements in ...
0
votes
1answer
47 views

Strange derivative

In this proof: http://www.math.hmc.edu/calculus/tutorials/mean_value/proof_mean.html Why does $g'(x) = f'(x) - \frac{f(b)-f(a)}{b-a}$?
1
vote
1answer
22 views

Partial Order proof with operation on a set

Let X be a set and let $f$ be an operation on X (i.e. it is a function from X $\times$ X to X), which we will denote with $f(x, y) = xy$. In addition, $x \le y$ iff $f(x, y) = x$. Suppose further that ...
2
votes
1answer
43 views

Centralizer Proof: $A \subseteq C(C(A))$

Show that $ A \subseteq C(C(A))$ Let $G$ be a group and $ A \subseteq G $. The centralizer of a subset of A is the set $C(A)=\{x\in G : ax=xa$ for all $a \in A\}$. *Isn't this obvious because $A=A$ ...
3
votes
2answers
235 views

Beautiful proof for $e^{i \pi} = -1$ [closed]

To celebrate the recent neuroscientific study that shows the beauty of math is in the mind, what is your most beautiful proof that $e^{i \pi} = -1$?
3
votes
1answer
94 views

Maximal total-weight matching in bipartite graph problem

Given a $G(A,B,E)$ bipartite graph and a $w: E \to R$ weight function. Problem 1: We are looking for a $M$ matching where the sum of the weights of edges in the M matching is maximal. This problem ...
1
vote
1answer
71 views

What is the proper way to format a hypothetical syllogism proof?

Problem: Show that these three statements are equivalent, where $a, b \in R:$ (i) $a < b$, (ii) the average of $a, b,$ is greater than $a,$ and (iii) the average of $a$ and $b$ is less than $b$. ...
2
votes
0answers
27 views

Selecting a unique pair satisfying a condition $\varphi$ with an ordering

Given a finite structure $\mathfrak{A}$ with Universe $|A| < \infty$ and signature $\tau$. We say a pair $(a,a') \in A$ satisfies a $\tau$-formular $\varphi$ iff $$ \mathfrak{A} \models ...
2
votes
1answer
162 views

Combinatorial Proof -$\ n \choose r $ = $\frac nr$$\ n-1 \choose r-1$

I'm reading about combinatorics, specifically 'Cohen's Introduction to Combinatorial Theory', and am stuck on one of the problems. I'm looking for a combinatorial proof for the following : $\ n ...
0
votes
2answers
226 views

For all x there exists a y such that x+y=0

I know this statement is true but I am having trouble actually proving it. I know that if x=5 then y=-5. How can you prove that properly.
0
votes
1answer
37 views

$728|a^{27}-a^3$ for all $a \in \mathbb Z$ [closed]

Proof that $728|a^{27}-a^3$ for all $a \in \mathbb Z$. I can't seem to find how to do this, any help?