Questions tagged [alternative-proof]
If you already have a proof for some result, but want to ask for a different proof (using different methods).
1
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1answer
33 views
Determine where this function is continuous - floor function
Determine where this function is continuous:
$$f(x)=
\begin{cases}
\frac{1}{x-\lfloor x\rfloor}, & \text{for } x \notin \mathbb{Z}\\
1, & \text{for } x \in \mathbb{Z}\\
\end{cases} $$
...
1
vote
3answers
32 views
Proving that a line segment joining the midpoints of two sides of a triangle equals half of third side.
Is this proof correct? I know it can be proven using the Midpoint Theorem. I believe the proof is correct, in that case, please explain how I should prove it with the same approach without a graph. If ...
1
vote
4answers
44 views
How many triplet primes of the form $p, p+2, p+4$ are there? Prove your conjecture.
I am giving this problem to 8th grade students, and I am hoping that people can help me find elementary ways to prove this problem. I would love to find other arguments that are accessible to 8th ...
2
votes
0answers
33 views
Another proof for impossibility of covering $\mathbb{R}^{n}$ with a set of varieties of cardinality less than $2^{\aleph_0}$
Let $\mathbb{R}^{n}$ be the affine $n$-space. Let $\{V_{\alpha} \}_{\alpha \in \Gamma}$ be a set of real varieteis such that $\mathbb{R}^{n} \subseteq {\bigcup V_{\alpha}}.$ Prove that $|\Gamma| \geq ...
4
votes
1answer
33 views
Another proof for $\lim\limits_{n \to \infty}n^{\frac{1}{n}}=1(n=1,2,\cdots)$. [closed]
Problem
Prove$$\lim\limits_{n \to \infty}n^{\frac{1}{n}}=1$$where $n=1,2,\cdots$
Proof
Notice that $$1=1^{\frac{1}{n}}\leq n^{\frac{1}{n}}=\sqrt[n]{1\cdot 1\cdot 1\cdots\sqrt{n}\cdot \sqrt{n}}\leq \...
2
votes
1answer
86 views
Find condition for $\alpha + \beta+ \gamma=\alpha . \beta. \gamma$
Find condition for real numbers $\alpha , \beta, \gamma$ such that $\alpha + \beta+ \gamma=\alpha . \beta. \gamma$
Solution:
we have this set for example:
$\sqrt 3+\sqrt 3+\sqrt 3=\sqrt 3.\sqrt 3.\...
0
votes
1answer
93 views
The number of permutations of $\mathbb{N}$
Let $\mathbb{N}!$ be the set of permutations of the natural numbers.
With $\aleph_0 = \text{card}\ \mathbb{N}$ you need two facts of cardinal arithmetic to show that the number of permutations of $ \...
0
votes
1answer
21 views
A second approach to a proof of the External Angle Theorem for triangles
Hello, I have been given the task shown to solve. I have done so. I am now trying to find a second way of solving thus being able to anticipate any answer someone may have when solving.
I was able ...
0
votes
1answer
17 views
Intuition about combining half-open intervals into inequality with absolute value?
I'm think about how to use the two conditions
$$0\le r\lt n\\
0\le r'\lt n,$$
to prove $\lvert r'-r\rvert\lt n,$ and the way I achieved this is by backwardly expand the result into
$$-n\lt(r'-r)\lt ...
1
vote
2answers
63 views
Prove that 3,5,7 are only 3 consecutive odd primes. [duplicate]
I have done a number theory proof. Showing that if a prime $p > 5$ then atleast one of $p-2,p+2$ is a multiple of 3. And when $p=3$ then that multiple is 3 itself.
However this question was ...
3
votes
1answer
37 views
Groups are cocomplete by adjoint functor theorem without explicit use of free products
$$
\newcommand{GRP}{\mathsf{GRP}}
\newcommand{I}{\mathcal{I}}
\newcommand{im}{\mathrm{Im} \;}
$$
This is a follow up for the question Royal Road to Free Groups and Free Products. I was exploiring ...
2
votes
1answer
52 views
An elementary proof of $\pi_1(S^n) =1$ using this general theorem
I was looking for an elementary proof of $ S^n$ being simply connected. My favourite is the one given by the user Olivier Bégassat in this post.
Then i realized that what he did is a particular case ...
-2
votes
3answers
75 views
Prove that if $f(x)=0$ except for a finite number of points, then $f\in\mathcal{R}[a,b]$ and that $\int^{b}_{a}f(x)dx=0.$ [closed]
Suppose $f:[a,b]\to\Bbb{R}$ and that $f(x)=0$ except for a finite number of points $c_1,c_2,\cdots,c_n$ in $[a,b]$. Prove that $f\in\mathcal{R}[a,b]$ and that $\int^{b}_{a}f(x)dx=0.$
My pdf presents ...
0
votes
0answers
21 views
Inversion in a sphere preserves circles (proof)
Inversion in the unit sphere, for a vector $x$, is defined by
$$\frac1x = \frac x{x^2} = \Big(\frac1{x\cdot x}\Big)x$$
How can we prove that a circle's inversion in the sphere is also a circle?
(I ...
2
votes
2answers
108 views
Dynamical systems proofs wanted
Recently I read dynamical systems proofs of the irrationality of $\sqrt{2}$ and Fermat's theorem. Now, I'm interested in other dynamical systems proofs of well-known things similar to this. What book, ...
3
votes
2answers
37 views
Alternative proof for subbasis for order topology and product topology: Finite intersections of elements of $\mathscr S$ is a basis
Munkres Topology: Both of the following proofs are at the very end of, respectively, Sections 14 and 15, which seem to correspond with the definition of a subbasis' being at the very end of Section 13....
2
votes
2answers
97 views
Prove by induction that $\sum_{n=1}^\infty \frac{1}{2^n} = 1$
The title explains the problem fairly well; is there a way to prove by induction that $\sum_{n=1}^\infty \frac{1}{2^n} = 1$. If not are there other ways?
I have thought of showing it by rewriting ...
2
votes
0answers
20 views
Does this prove the Inscribed Rectangle/Square Theorem?
Imagine a circle, with four points on it forming a rectangle.
Twisting and bending this circle along those four points, any curve can be produced. $\blacksquare$
Is this a proof of the Inscribed ...
0
votes
1answer
42 views
Calculus Circle Related rates circle proof
NOTE: THIS QUESTION HAS ALREADY BEEN ANSWERED AND CAN BE FOUND AT THIS LINK (Related rates circle problem)
Question:
Two circles $A$ and $B$ have the same center. The radius of the inner circle $A$ ...
0
votes
2answers
78 views
proof that $[0,1]$ is compact
I tried coming up with a proof of compactness of $[0,1]$ in $\mathbb{R}$ and thought of the following method. please let me know if it is correct or how it could be made more correct.
For any open ...
0
votes
1answer
35 views
Alternate way to show these two integrals are the same
Let $T>0$ and $a,b,x\in \mathbb{R}$ such that $a<b$.
Take $I:=\int_{-T}^T \frac{e^{-it(x-b)} - e^{-it(x-a)}}{-it} dt$.
Take $J:=2 \int_{x-b}^{x-a}\frac{\sin(Tu)}{u}du$.
One way to show $I=J$ ...
0
votes
1answer
25 views
“Every convergent sequence is bounded” and the choice of epsilon
In the proof of the theorem which states that every convergent sequence is bounded, it's chosen $\varepsilon=1>0$, but the proof works for every $\varepsilon>0$ right?
Couldn't the proof be ...
0
votes
0answers
73 views
Is there a combinatorial proof of this recurrence relation for families of sets thus defined?
Define $S_{k,n}$ like in this question. I think I have proved the identity $$\sum_{m\in S_{k,n+1}}m^{-1}=\sum_{m\in S_{k,n}}m^{-1}+\frac1{p_{n+1}}\sum_{m\in S_{k-1,n}}m^{-1} $$for all $n\ge k\ge1$. In ...
0
votes
0answers
26 views
What are elementary proofs in the context of geometry and arithmetic?
In his answer to the question Is an integer a sum of two rational squares iff it is a sum of two integer squares? André Nicolas mentions that the proof
that if $m$ is a positive integer, then $m$ ...
5
votes
1answer
26 views
Congruence equation for distinct primes $p,q$.
Prove that, for distinct primes $p,q$: $$\tag{1} p^{q-1} + q^{p-1} \equiv{1} \pmod{pq}.$$
This is a pretty straight forward problem, I know, and although the standard proof is simpler, I was ...
1
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0answers
24 views
Geometrical proof of the impossibility of angle trisection by straightedge and compass
There's a fascinating (almost) geometrical proof of the impossibility of angle trisection by straightedge and compass by Terrence Tao which is somehow more comprehensible (and more directly to the ...
0
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1answer
29 views
Proof of Laplace Transform of *Real Number* Powers
As I understand it, the theorem and proof for the Laplace transform of positive integer powers is as follows:
Theorem
Let $t^n : \mathbb{R} \to \mathbb{R}$ be $t$ to the $n$th power for some $...
0
votes
1answer
26 views
Alternative proof for chains
Let $(X,\le)$ be a chain, $S\subseteq X$, and suppose $\sigma=\sup S$ exists. Then for every $x\in X$ with $x<\sigma$, there is some $s\in S$ s.t. $x<s\le\sigma$.
By contradiction: Let $x\in X$ ...
3
votes
3answers
145 views
How show that $\sum_{j=0}^\infty \frac{(-1)^j\binom{\frac12}{j}}{{2j+1}}=\frac{\pi}4$
As discussed in detail in that OP Find the limit of $\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ \frac { \sqrt { { n }^{ 2 } - { k }^{ 2 } } } { { n }^{ 2 } } } } $, using Riemann sum we ...
5
votes
6answers
146 views
What is the relation between 'the order of $x^k = n/{\gcd(k,n)}$' and Lagrange's Theorem?
Algebra by Michael Artin Cor 2.8.11 to Lagrange's Theorem (Theorem 2.8.9).
Question: What is the relation between Cor 2.8.11 and the order of $x^k$ given by $n/{\gcd(k,n)}$ (in the text, ...
2
votes
2answers
83 views
Proof verification $(x_1^2 + x_2^2)(y_1^2 + y_2^2) = (x_1y_1 + x_2y_2)^2+ (x_1y_2 - x_2y_1)^2$ (Spivak's Calculus)
I have to prove the Schwarz inequality with that, then I want to know if what I did was so rigorous or not (And if it's good).
$$(x_1^2 + x_2^2)(y_1^2 + y_2^2) = (x_1y_1 + x_2y_2)^2 + (x_1y_2 - ...
5
votes
1answer
132 views
How would Euclid have proved that $ a \times b = ((ab)/c) \times c$?
The following statement seems too obvious to prove or even to mention:
That the area of a rectangle with side lengths $a$ and $b$ equals the area of a rectangle with side lengths $(ab)/c$ and $c$ for ...
2
votes
1answer
38 views
Determining whether proofs of the Pythagorean theorem are essentially the same, or essentially different
There are great annotated lists of proofs of the Pythagorean theorem, to name just two:
cut-the-knot.org
brilliant.org
There's an obvious difference between geometric and algebraic proofs (which ...
1
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0answers
22 views
Alternative proof of Levy's continuity theorem(problem with understanding specific part)
I found alternative proof of Levy's continuity theorem. In spite of reading this proof 12 times and being able to find explanation to therse parts, which were not clear for me, there is 1 crucial ...
0
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2answers
28 views
Proof of a practical method of a natural number's equation
Example Question :
$3m + 4n = 70$, $m,n$ are natural numbers. How many values can $m$ have?
I learned a method to solve this kind of problem, but I've never thought about that before.
for
n=1 =...
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votes
5answers
112 views
Prove that $1+x<e^x$ for all real numbers $x\neq0$.
Prove that $1+x<e^x$ for all real numbers $x\neq0$.
Here is my solution:
We'll prove this as two cases; first when $x<0$ and then when $x>0$.
Let $\alpha>0$. Then,
$$e^\alpha-\alpha e^\...
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1answer
96 views
Prove cyclic subgroup has n distinct elements $\langle x \rangle = \{1,x,x^2,..,x^{n-1}\}$ without using Euclidean algorithm again
Algebra by Michael Artin Prop 2.4.2
Here is the statement and the proof:
I split Prop 2.4.2(c) into Parts I, II and III respectively:
Part I: There is some positive integer $n$ s.t. $S=\mathbb ...
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votes
1answer
52 views
Questions on the length of a positive integer $a\in \mathbb{Z}^+$.
Definition. We call length of a positive integer $a\in \mathbb{Z}^+$, and we write $\ell(a)$, the number of $2$-adic digits
of $a$. That is $\ell(a):=\lfloor \log _2 a\rfloor +1$.
I found on my ...
1
vote
2answers
51 views
Proof about a theorem that says a function is continuous if only if f is right continuous in a and left continuous.
Theorem:
A function $f:D\to \mathbb R$ is continuous in $a\in D$ $\iff f$ is left and right continuous in $a$.
Proof:
I firstly thought just to write down the definitions of left and right ...
0
votes
0answers
75 views
Riemann surface of $f(z)=\sqrt{z^2+1}$
Construct the Riemann surface of $f(z)=\sqrt{z^2+1}$.
Notice that $f(z)=\sqrt{z^2+1}=\sqrt{z+i}\sqrt{z-i}$.
Define $f_1(z)=\sqrt{r_1r_2}e^{\frac{\theta_1+\theta_2}{2}}$ where $r_1=\vert z-i\vert,r_2=...
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1answer
80 views
Proof explanation: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ [duplicate]
Two days ago, I asked a question
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ but was answered just once. However, I am finding it ...
1
vote
1answer
40 views
Convergence of a logarithmic sequence
Is the sequence $\left\{\ln\left((1+\frac1n)^n\right)\right\}_{n=1}^{\infty}$ convergent or divergent?.
I tried to solve it by L Hospital's rule and arrived at 0...implying it is convergent..is it?
...
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votes
0answers
6 views
Trying to prove BIBO stability implies the boundedness of the impulse response of an LTI system without a counterexample
I have tried to prove that BIBO stability implies that the impulse response of the LTI system, can someone please check if the proof is correct?Proof part 1/2
proof part2/2
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0answers
41 views
Riemann Integrability of a function $f$ on $[0,1]$
The question goes like this:
Suppose is a bounded function such that $f$ is Riemann integrable on $[a,1]$ for every $a \in (0,1)$. Is $f$ Riemann integrale on $[0,1]$?
Duplicate
My approach:
...
4
votes
6answers
276 views
Can we prove that $\lim_{n\to\infty} \frac{1^{n_0}+2^{n_0}+\cdots+n^{n_0}}{n^{n_0+1}}$ is finite for any $n_0\in\Bbb N$ without a direct computation?
Can we prove without direct calculation that this limit is finite for any natural number $n_0 \in \mathbb{N}$?
$$ \lim_{n \to \infty} \frac{1^{n_0}+2^{n_0}+\cdots+n^{n_0}}{n^{n_0+1}} $$
2
votes
1answer
55 views
Application of Implicit Function Theorem: Problem 2-40 from Spivak's Calculus on Manifolds
At the end of the section on Implicit Functions in Spivak's Calculus on Manifolds, we have Problem 2-40:
Problem 2-40. Use the implicit function theorem to re-do Problem 2-15(c).
And, Problem 2-15(...
0
votes
0answers
32 views
Is there any proof of cauchy formula for repeated integration without induction?
https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration
Here, the formula has been proved by using mathematical induction. Is there any other way to prove it? I mean how did cauchy ...
1
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0answers
45 views
Let $a,b\in\mathbb{R}^* = [-\infty,\infty]$ be such that $a\leq b + \varepsilon$ for all $\varepsilon>0$. Show $a\leq b$.
I wish to show the above statement. I've shown this to be true by taking cases of $a$ and $b$.
For $a,b\in\mathbb{R}$, I assumed $a>b$ and derived a contradiction.
The cases involving $\pm \...
0
votes
2answers
37 views
Differentiation of solution to time-dependent system of equations: Problem 2-15(c) from Spivak's Calculus on Manifolds
This is Problem 2-15(c) from Spivak's Calculus on Manifolds.
Problem 2-15(c). If $\det(a_{ij}(t)) \neq 0$ for all $t$ and $b_1,\dots,b_n : \mathbb{R} \to \mathbb{R}$ are differentiable, let $s_1,\...
1
vote
1answer
56 views
Matrices of rank 1 and the inequality $rank(A+B) \leq rank(A) + rank(B)$
I'm interested in a proof of the inequality $rank(A+B) \leq rank(A) + rank(B)$ that does not use the bases of $A$'s and $B$'s row/column spaces, as described in other answers on this site. My idea is ...
