A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
Hello!
I have some troubles diving in the proof of this lemma
Lemma. Let ##S## be locally compact, Hausdorff and second countable. Then every open cover ##\lbrace U_\alpha \rbrace## of ##S## has a countable, locally finite refinement consisting of open sets with compact closures.
Proof...
Proving this geometrically [1] gives ##J = r.##
Why is the ##-r## one wrong? Why is ##(x, y) \rightarrow (\theta, r)## is different from ##(x, y) \rightarrow (r, \theta)##? Edit: In Paul's Notes [2] it seems like ##J## is always positive, but online says it can be negative...
[1] The first...
I'm reading "Complex Made Simple" by David C. Ullrich and here i have a problem with the proof of a theorem:
Theorem
Suppose that ##p : X \to Y## is a covering map. If ##\gamma : [0,1] \to Y## is continuous, ##x_0 \in X## and ##p(x_0) = \gamma(0)## then there exists a unique continuous function...
In the photos are two proof questions requiring proving convergence of sequence from convergent subsequences. Are my proofs for these two questions correct? Note in the first question I have already proved that f_n_k is both monotone and bounded
Thanks a lot in advance!
For this problem,
My proof is
Since ##f'## is increasing then ##x < y <z## which then ##f(x) < f(y) < f(z)##
This is because,
##f''(t) \ge 0## for all t
## \rightarrow \int \frac{df'}{dt} dt \ge \int 0~dt = 0## for all t
##\rightarrow \int df' \geq 0## for all t
##f ' \geq 0## for all t...
For this problem,
I'm confused by the implication from the antecedent ##0 < |x - c| < \delta## to the consequent. Should the consequent not be ##|f''(x) - f''(c)| < \frac{1}{2}## where ##\epsilon = \frac{1}{2}## (Since we are applying the definition of a limit for the first derivative curve)...
For this problem,
My solution is
If ##c < 1##, then let a be a number such that ##c < a < 1 \implies c < a##. Thus for some natural number such that ##n \geq N##
##|x_n|^{\frac{1}{n}} < a## is the same as ## |x_n| < a^n##
By Law of Algebra, one can take the summation of both sides to get...
For this problem,
My solution is,
##F(x)=\left\{\begin{array}{ll} e^{-\frac{1}{x}} & \text { if } x>0 \\ 0 & \text { if } x \leq 0\end{array}\right.##
The we differentiate both sub-function of the piecewise function. Note I assume differentiable since we are proving a result that the function...
If ##V## is timelike Killing with Frobenius condition ##V_{[\alpha} \nabla_{\mu} V_{\nu]} = 0## then you can derive the equation:$$\nabla_{\mu} (|V|^2 V_{\nu}) - \nabla_{\nu} (|V|^2 V_{\mu}) = 0$$which has the solution$$V_{\alpha} = \partial_{\alpha} \phi \quad \mathrm{where} \quad \phi = x^0 +...
Hi,
I don't know if I have solved task correctly
I used the epsilon-delta definition for the proof, so it must hold for ##f,g \in (C^0(I), \| \cdot \|_I)## ##\sup_{x \in [a,b]} |F(x)-G(x)|< \delta \longrightarrow \quad |\int_{a}^{b} f(x)dx - \int_{a}^{b} g(x)dx |< \epsilon##
I then...
For this true or false problem,
My solution is,
With rearrangement ##\frac{f(x) - f(a)}{x - a} > f'(a)## for ##x < a## since ##f''(x) > 0## implies ##f'(x)) > 0## from integration. ##f'(x) > 0## is equivalent to ##f(x)## is strictly increase which means that ##\frac{f(x) - f(a)}{x - a} > f'(a)...
For this problem,
I am trying to prove that this function is non-differentiable at 0.
In order for a function to be non-differentiable at zero, then the derivative must not exist at zero ##⇔ \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0}## does not exist or ##⇔ \lim_{x \to 0^-} \frac{f(x) - f(0)}{x...
For this problem,
THe solution is,
However, does someone please know why from this step ##-1 \leq \cos(\frac{1}{x}) \leq 1## they don't just do ##-x \leq x\cos(\frac{1}{x}) \leq x## from multiplying both sides by the monomial linear function ##x##
##\lim_{x \to 0} - x = \lim_{x \to 0} x= 0##...
We consider base case (##n = 1##), ##B\vec x = \alpha \vec x##, this is true, so base case holds.
Now consider case ##n = 2##, then ##B^2\vec x = B(B\vec x) = B(\alpha \vec x) = \alpha(B\vec x) = \alpha(\alpha \vec x) = \alpha^2 \vec x##
Now consider ##n = m## case,
##B^m\vec x = B(B^{m - 1}...
I have a doubt about this problem.
(a) Show that a matrix ##\left(\begin{array}{ll}e & g \\ 0 & f\end{array}\right)## has determinant equal to the product of the elements on the leading diagonal. Can you generalize this idea to any ##n \times n## matrix? The first part is simple, it is just ef...
I am trying to solve (a) and (b) of this tutorial question.
(a) Attempt:
Since ##c'## is at ##c'(0) = 1##, then from the definition of continuity at a point:
Let ##\epsilon > 0##, then there exists ##d > 0## such that ##|x - 0| < d \implies |c'(x) - c'(0)| < \epsilon## which is equivalent to...
Consider this proof:
Is it a valid proof?
When we divide by ##z##, we assume that ##z \neq 0##. So, we cannot put ##z=0## on the next step. IOW, after dividing by ##z## we only know that $$c_1+c_2z+c_3z^2+...=d_1+d_2z+d_3z^2+...$$ in a neighborhood of ##0## excluding ##0##.
For this problem,
The solution is,
However, does someone please know why this did not use ##2n ≤ 2n^2 + 2n + 1## which would give
##\frac{3n - 1}{2n^2 + 2n + 1} ≤ \frac{3n}{2n} = \frac{3}{2}##?
In general, after solving many problems, it seems that when proving the convergence of a rational...
The problem and solution are,
However, I am confused how the separation vector between the two masses is
##\vec x = x \hat{k} = x_2 \hat{x_2} - x_1 \hat{x_1}= l\theta_2 \hat{x_2} - l\theta_1 \hat{x_1 } = l(\theta_2 - \theta_1) \hat{k}##. where I define the unit vector from mass 2 to mass 1...
I am trying to understand the proof given in Ethan Bloch's book "The real numbers and real analysis". I am posting snapshot of the proof in the book.
I am also posting theorem 1.2.9 given in the book.
Here author is trying proof by contradiction. First, I don't understand why specific...
Hello everyone,
I've been trying to understand the proof for the binomial theorem and have been using this inductive proof for understanding.
So far the proof seems consistent everywhere it's explicit with the pattern it states, but I've started wondering if I actually fully grock it because I...
Hello, found this proof online, I was wondering why they defined r_2=r_1-(r_1^2-2)/(r_1+2)? i understand the numerator, because if i did r_1^2-4 then there might be a chance that this becomes negative. But for the denominator, instead of plus 2, can i do plus 10 as well? or some other number...
My first solution is
Let
##S = \{x_1, x_2, x_3, ..., x_n\}##
##T = \{2x_1, 2x_2, 2x_3, ... 2x_n\}##
##T = 2S##
Therefore, ##inf T = inf 2S = 2inf S = 2M##
May someone please know whether this counts as a proof?
My second solution is,
##x ≥ M##
##2x ≥ 2M##
##y ≥ 2M## (Letting y = 2M)
Let...
For this problem,
My solution:
Using definition of Supremum,
(a) ##M ≥ s## for all s
(b) ## K ≥ s## for all s implying ##K ≥ M##
##M ≥ s##
##M + \epsilon ≥ s + \epsilon##
##K ≥ s + \epsilon## (Defintion of upper bound)
##K ≥ M ≥ s + \epsilon## (b) in definition of Supremum
##M ≥ s +...
Hello everyone, I've been brushing up on some calculus and had some new questions come to mind.
I notice that most proofs of the fundamental theorem of calculus (the one stating the derivative of the accumulation function of f is equal to f itself) only use a limit where the derivative is...
I've read a proof from Complex Made Simple (David C. Ullrich)
Proposition 4.3. Suppose that ##V## is an open subset of the plane. There exists a branch of the logarithm in ##V## if and only if there exists ##f \in H(V)## with ##f'(z) = \frac{1}{z}## for all ##z \in V##.
Proof: One direction is...
Hello everyone,
maybe some of you know the formula for the number of multiplications in the FFT algorithm. This is again given as ##N/2 \cdot log(N)##. Why is that so? Can you really "prove" this?
I can only deduce this from what I know, because we have ##log(N)## levels and ##N/2##...
Hello everyone,
I found a good proof for the area of a circle being ##{\pi}r^2## but I was recently working on my own proof and I used a change of variables and was wondering if I did it correctly and why a change of variables seems to work.
I start with the equation of a circle ##r^2 = x^2 +...
Hi people,
It's been years I wanted to post this question here. I would like to build a zero knowledge proof that a given chess position contains at least one checkmate. I know that anything provable admits a zero k proof. I know about...
Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book where is proved but I failed. This is the theorem:
Let
$$ x_0 < x_1 < \cdots < x_n \in [a, b]...
I'm looking for theorems related to using modulo arithmetic.
As an example, if I apply a sequence of arithmetic operations to a given number to get an answer and then apply a modulo operation on the result to get a remainder in a given base. Wiil that be the same if I apply the modulo operation...
I've tried this problem so, so, so so so many times. Given the equations above, the proof starts easily enough:
$$\partial_\mu T^{\mu\nu}=\partial_\mu (∂^μ ϕ∂^ν ϕ)-\eta^{\mu\nu}\partial_\mu[\frac{1}{2}∂^2ϕ−\frac{1}{2}m^2ϕ^2]$$
apply product rule to all terms
$$=\partial^\nu \phi \cdot...
I tried to prove this but I fall into a loop when I try to apply integration by factors, that is I prove that the integral is equal to itself.
Any helpfull tips?
Want to understand how set C contains ##N## x H. H is only defined to be a set with element e and as the domain/range of function k. Is this enough information to conclude that the second set in the cartesian product W is H and not a subset of H?
My thinking is to show that ##N## and H satisfy...
"bubbles are ball" is called isoperimetric problem in serious mathematic. In this topic, many essay were written. Here's my serious essay about "why earth ball", which has been rejected by arxiv and my mentors...... I would want to know if physicists are interest?
I really think that is...
Hi,
I have problems proving task d
I then started with task c and rewrote it as follows ##\lim_{n\to\infty}\sum\limits_{k=0}^{N}\Bigl( \frac{z^k}{k!} - \binom{n}{k} \frac{z^k}{n^k} \Bigr)=0 \quad \rightarrow \quad \lim_{n\to\infty}\sum\limits_{k=0}^{N} \frac{z^k}{k!} =...
with this background, we proceed to the proof. Let us define a set
$$ G = \{ z \in \mathbb{N} | \; x, y \in \mathbb{N}\; (x \cdot y) \cdot z = x \cdot (y \cdot z) \} $$
We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above...
with this background, we proceed to the proof. Let us define a set
$$ G = \{ z \in \mathbb{N} | \mbox{ if } y \in \mathbb{N}, y\cdot z = z \cdot y \} $$
We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above. Obviously, ## G...
I want to prove that ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates where ##a,b,c \in \mathbb{N}##.
The book I am using ("The real numbers and real analysis" by Ethan Bloch ) defines Peano postulates little differently.
Following is a set of Peano postulates I am using. (Axiom 1.2.1...
I have to prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##.
The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently.
Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book)
There exists a set...
I have to prove that ##1 + a = s(a) = a + 1## using Peano postulates if ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently.
Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book)...
Hi
I have to prove the following three tasks
I now wanted to prove three tasks with a direct proof, e.g. for task a)$$\sqrt[n]{n} = n^{\frac{1}{n}}= e^{ln(n^{\frac{1}{n}})}=e^{\frac{1}{n}ln(n)}$$
$$\displaystyle{\lim_{n \to \infty}} \sqrt[n]{n}= \displaystyle{\lim_{n \to \infty}}...
Hi,
I'm having problems with the proof for the induction of the following problem: ##\sum\limits_{k=0}^{n} \frac{(-1)^k}{k+1} \binom{n}{k}=\frac{1}{n+1}## with ##n \in \mathbb{N}##
I proceeded as follows:
$$\sum\limits_{k=0}^{n+1} \frac{(-1)^k}{k+1} \binom{n+1}{k}=\frac{1}{n+2}$$...
Hi,
I have problems with the proof for task a
I started with the supremum first, but the proof for the infimum would go the same way. I used an epsilon neighborhood for the proof
I then argued as follows that for ##b- \epsilon## the following holds ##b- \epsilon < b## and ##b- \epsilon \in...
I let,
## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}##
##\tan^{-1}\left[\dfrac{1}{5}\right]- \dfrac{1}{4}\tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{16}##
Then i let, ##\tan^{-1}\left[\dfrac{1}{5}\right] = α ...
Have I proved Fermat last theorem?
X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formular is fine. Set x=X^2, y=Y^2, z=Z^2, so x, y, z are (some) integer numbers based on X,Y,Z.
x^4 + y^4 != z^4 //x, y, z are still integer, would be obey to Fermat's Fermat...