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.

learn more… | top users | synonyms

1
vote
0answers
115 views

So we don't need to choose delta, epsilon, or $N \in \mathbb{N}$ in delta-epsilon or sequence convergence proofs?

(http://math.stackexchange.com/a/700667/85079) I would write the proof with all my bounds $\eta$ and then choose $\eta$ to make the conclusion match the arbitrary $\epsilon$. ...
-1
votes
1answer
40 views

Prove: $a < a^n$ (more details in description)

Let $\rm\:a\in \mathbb Z.\:$ Prove that if $\rm\: a > 1,$ then for all $\rm\:n > 1, a < a^n.$
0
votes
2answers
74 views

Prove that there is a real solution of $x=e^{-x}$

I know I have to use the intermediate value theorem but how?
4
votes
4answers
67 views

Prove that $a \equiv b \pmod{m_1m_2}\implies a \equiv b \pmod {m_1}$

So, I have this problem: if $$a \equiv b \mod(m_1m_2)$$ then (show) $$a \equiv b \mod(m_1)$$. I have to do a proof, but I have no idea where to begin the proof. Can someone help? Proof (Edit): ...
3
votes
2answers
154 views

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)
0
votes
1answer
81 views

Differential map on the vector space of polynomials: Kernel and Image

Given the $V_n$ is the vector space of polynomials of degree $\leq$ n over $\Bbb R$ So $M_D = \begin{bmatrix} 0 & 1 & ... & 0\\ 0 & 0 & 2 &... 0 \\ 0 & 0 &0 &.\\ . ...
3
votes
3answers
118 views

When does one proof of one direction of an If and Only If proof suffice?

Would someone please explain when this is admissible (please expound on $\color{darkred}{sometimes}$)? In advance of starting an Iff proof, how would one divine/previse if this convenience (of a ...
5
votes
7answers
248 views

Within If and Only If Proofs, why can the proof for one direction be easier than the other?

For $ P \iff Q$, my initial sentiment is that because P and Q are equivalent, the total of two proofs (one for each direction) should entail the equivalent level of "difficulty" or "exertion", as well ...
1
vote
2answers
44 views

Prove by induction on a string

I want to practice proving the following: Given a binary string s, I want to prove $s$ has the same number of substrings 01 and 10 $\iff$ the first and last character of $s$ is the same. For ...
0
votes
2answers
24 views

proof by induction - creating summations?

I have two proofs I need to do that I can not figure out how to turn into summations in order to solve. $3|(4^n-1)$ I believe that $|$ is meant to symbolize $3$ divides ... $n!\le n^n$ I have to ...
1
vote
3answers
35 views

EXCERSICE VERIFICATION: Find where $f(x):=|x|+|x+1|$ is differentiable and calculate its derivative

Could someone verify my excersice? a) $f(x):=|x|+|x+1|$ First, analyse the roots of each absolute value, where they go to zero: $$|x|:=\left\{\begin{matrix} & x& x>0 \\ & x- ...
0
votes
0answers
63 views

Finding the coefficients of a partial differential equation after a change of coordinates.

I'm stuck in one of the mathematical steps of my physical problem. I've been following the derivation of my equations (starting at section 4) from this article Symmetric Euler-Angle Decomposition of ...
4
votes
2answers
124 views

When do we write “we are done”?

This may seem like a bit of a silly question, but I notice that in some proofs (a remarkable amount), the author writes: "We are done." after completing a proof. Is this the equivalent of writing one ...
1
vote
3answers
49 views

Prove that for all $x$ where $0<x<\pi/2$, $\sin x+\cos x>1$

Prove that for all $x$ where $0<x<\pi/2$, $$\sin x + \cos x > 1.$$ I tried multiple Identities I do not know what I am missing. I have tried changing into different identities.
1
vote
3answers
210 views

Probability Proof $ P[(A \cap B^c) \cup (A^c \cap B)] = P(A) + P(B) - 2P( A \cap B) $

How would I go about proving this statement: $ P[(A \cap B^c) \cup (A^c \cap B)] = P(A) + P(B) - 2P( A \cap B) $ Describe in English the event where the probability is computed by the expression on ...
0
votes
2answers
29 views

Using the method of induction

Can someone help solve this problem? Prove that if $n≥1$ and $a_1,a_2,….,a_n$ are any real numbers, then $|a_1+a_2+⋯+a_n |≤|a_1 |+|a_2 |+⋯+|a_n |$.
0
votes
1answer
114 views

Comparison Theorem for Integrals

Problem: Let $a>0$ and $b>a+1$. Use the Comparison Theorem to show that the following integral is convergent: $$\int ^ \infty _0 \frac{x^a}{1+x^b} \ dx$$ My attempt at this was that since ...
0
votes
3answers
60 views

Help with proof of continuous functions with neighborhood value $N$.

Prove that if $f(x)$ and $g(x)$ are continuous at $c$ and $f(c) < g(c)$ then there is a neighborhood $N$ of $c$ such that $f(x) < g(x)$.
0
votes
1answer
21 views

Name for proof by logical equivalence

A discussion on ELU stackexchange has led to the question of whether there is a name for the style of proof in which you start with the proposition to be proven and then proceed via a chain of ...
16
votes
3answers
747 views

Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based. To what extent is it best not to start a sentence with a mathematical symbol? I find that when trying to solve a problem or ...
4
votes
3answers
273 views

Do you need to find the domain of a function to prove injectivity?

I teach math and have had this debate with a fellow teacher. I believe the domain of a function is really not important to determine if a function is or is not injective if there is no "real life" ...
4
votes
3answers
225 views

Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
1
vote
3answers
68 views

Wrting equations for work rate problems

Consider the following An experienced bricklayer can work twice as fast as an apprentice bricklayer. After the bricklayers work together on a job for 6 h, the experienced bricklayer quits. The ...
1
vote
0answers
54 views

Proof by induction for all positive integers

$\sum_{j=1}^n j^2 = \frac{n(n+1)(2n + 1)}{6}$ So I did the obvious and plugged one to show that $1^2 = \frac{1(1+1)(2(1) + 1)}{6}$ Now I am trying to show that $1^2 + 2^2 + ... + n^2 + (n + 1)^2 = ...
0
votes
0answers
40 views

Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
0
votes
1answer
49 views

If two matrices are similar, the geometric multiplicities of their eigenvalues are the same

Problem Let $A$ and $B$ be similar matrices. Prove that the geometric multiplicities of the eigenvalues of $A$ and $B$ are the same. [Hint: show that, if $B=P^{-1}AP$, then every eigenvector of ...
2
votes
1answer
167 views

Prove for every two sets $A$ and $B$

Prove for every two sets $A$ and $B$ that $A-B$, ${B-A}$ and $A \cap B$ are pairwise disjoint. I'm really stuck on this one. I know pairwise disjoint means no two elements in $A$ and $B$ are ...
0
votes
0answers
34 views

Proof sum of permutation

I'm trying to prove: $$P(N) = \sum permutation(A,N)=1 \tag{1}$$ for the particular choice of the set $A = \{ \mu_1, \dots, \mu_n, 1-\mu_1, \dots, 1-\mu_n \}$, where $i = 1, \dots, N$ . So for ...
0
votes
2answers
80 views

Prove 6 Divides n(n+1)(n+2) [closed]

Let n be an integer such that n >= 1. Prove that 6 divides n(n + 1)(n + 2). Not sure where to start, been stuck for a while
1
vote
0answers
47 views

Integral Domain and PID Proof

Prove that, in a domain, $(a)=(b)$ iff $a = bu$ for some unit $u$. By $(a)=(b)$, it also means that $a\mid b$ and $b\mid a$ so we can write them as $a=bu$ and $b=av$ for some $u, v \in R$ where $u$ ...
2
votes
1answer
47 views

Divisibility and Principal Ideal Domain Proof

Let R be a ring. Show that a|b iff $b \in (a)$ iff $(b) \subseteq $ (a). I first just want to write out what I know about this statement: a|b means that a divides b or a is divisible by b and there ...
0
votes
3answers
186 views

Maximal Proper Ideal is a field proof

Show that a proper ideal M of a commutative ring R is maximal if and only if R/M is a field. What I know: Because M is a proper ideal $M \neq R$. The ideal M is maximal if it is a maximal element ...
5
votes
1answer
272 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
2
votes
4answers
190 views

Proper Ideal and Units Proof

Show that an ideal of I R is proper if and only if it does not contain 1 iff and only if it does not contain any units. (1 is the identity element) I'll need to show 3 parts: (1) $\implies$ (2): ...
0
votes
0answers
44 views

Ideal Test Proof

Let $\emptyset \subset I \subseteq R$. Prove that I is an ideal of R if and only if $a-b, ra, ar$, $\in$ $I$ for all $a, b \in I$ and $r \in R$. I know that if I is an ideal in a ring R and $a \in ...
2
votes
1answer
75 views

Integral Domain and no nonzero divisors Proof

Prove that a commutative ring is an integral domain if and only if it has no nonzero zero divisors. I think my main problem is that I'm getting jumbled in the wording! By 'no nonzero zero divisors' ...
1
vote
1answer
35 views

Need help understanding a specific equality in this proof

Question. Let $f:\mathbb R\to \mathbb R$ be a uniformly continuous function. Show that there exists $a,b>0$ such that $|f(x)|\le a|x|+b,$ $\forall x\in\mathbb R$. Proof. Since $f$ is uniformly ...
0
votes
2answers
195 views

Prove that if M = supS for some nonempty bounded set S, then there exists an increasing sequence sn, of points S such that lim sn = M

Prove that if M = supS for some nonempty bounded set S, then there exists an increasing sequence sn, of points S such that lim sn = M Is this a monotone sequence? Do I need to use Cantor's principle
0
votes
3answers
59 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
65 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
57 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
59 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
97 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
214 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
822 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
143 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
29 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
94 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
179 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
59 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 $\mathbb Z{\sqrt{-2}}$. ...