For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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0answers
15 views

“Cascade induction”?

I refer to this answer. The answer is based on several simplification steps, all of them proven by induction. $$S_n = 2903^n - 803^n - 464^n + 261^n$$ $$T_n = 2642\cdot2903^n - 542\cdot803^n - ...
2
votes
3answers
28 views

Mistake in (Baby) Do Carmo? Elementary topology of surfaces.

If you have the book, it's proposition 2 of section 5.3. If not, the proposition reads: Given any two points p and q $\in$ a regular, connected surface S, there exists a parameterized piecewise ...
2
votes
1answer
11 views

Proving Cayley formula using Kirchhoff matrix theorem?

To count the number of spanning trees of a complete graph of order $n$ one can use Kirchhoff matrix theorem and arrive at the exact answer $n^{n-2}$. But in doing so, one should know how to evaluate ...
1
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2answers
21 views

Show $\sum_{1\leq n\leq x}\sum_{d\mid n}f(d)=\sum_{1\leq d\leq x}\sum_{1\leq m\leq x/d}f(d)$

I have been trying to get my head around this step in a proof, but havn't been able to, Question: Show $$\sum_{1\leq n\leq x}\sum_{d\mid n}f(d)=\sum_{1\leq d\leq x}\sum_{1\leq m\leq x/d}f(d)$$ ...
1
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1answer
23 views

Proof By Induction - $n^2 = \sum_{i=1} ^{n} (2i-1)$ for all $n\geq 1$

Using Proof By Induction I am trying to prove the following: $n^2 = \sum_{i=1} ^{n} (2i-1) $ for all $n\geq 1$ Here is my solutions so Far: Base Case: $n=1, LHS: 2(1)-1 = 1, RHS = 1^2 = 1, True$ ...
1
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4answers
26 views

Proving Two Sets are Equal - Infinite Sets - Example

Let $$A = \{x | x = 2n+1, n\in\mathbb{Z}\}$$ and $$B = \{x | x = 2m-21, m\in\mathbb{Z}\}.$$ I am trying to prove $A =B.$ I understand that I need to prove $A\subseteq B$ and $B\subseteq A$; But my ...
0
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1answer
15 views

Prove that $\int_{\mathbb{R}}x^me^{2ax}e^{-x^2/2}=e^{2a^2}\int_{\mathbb{R}}(x+2a)^me^{-x^2/2}$

As the title indicates, I've been trying for quite some time now to prove that $$\int_{\mathbb{R}}x^me^{2ax}e^{-x^2/2}=e^{2a^2}\int_{\mathbb{R}}(x+2a)^me^{-x^2/2}$$ $\forall m \in \mathbb{N}, \forall ...
3
votes
2answers
79 views

show that nth Chebyshev polynomial is an nth order polynomial

Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$. Show that $T_n(x)$ is an nth order polynomial This is my attempt, however I couldn't reduce it to a polynomial. ...
2
votes
1answer
21 views

Epsilon and Delta proof of $\lim_{x\to0} \frac{2-\sqrt{4-x}}{ x}$

I need to prove $\lim_{x\to0} \frac{2-\sqrt{4-x}}{ x}$ I first found the limit to be $\frac{1}{4}$ by using l'hopital's rule. By definition i need to find a $\delta > 0$ for every $\epsilon >0$ ...
0
votes
0answers
13 views

Orthogonal surfaces

Prove that the three surfaces of the family $xy/z=u$ $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}=v$, $\sqrt{x^2+y^2}-\sqrt{y^2+z^2}=w$ that pass through just one point are orthogonal I´m assuming that first I ...
5
votes
3answers
83 views

Where does the proof of $\sqrt 2$ is irrational break down when trying to prove the same for $\sqrt 4$?

I'm trying find where the common proof by contradiction that $\sqrt 2$ is irrational breaks down when trying to prove $\sqrt 4$ is irrational. Assume $(\frac pq)^2=4$ and $\gcd(p,q)=1$. I guess I ...
0
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3answers
42 views

Suppose $A$ is a subset of $B$ and $B$ is a subset of $C$ and $A$ is equinumerous with $C$. Prove $B$ is equinumerous with $C$.

Definition I use: $A \sim B$ means $A$ is equinumerous with $B$ which means there is a $f\colon A \rightarrow B$ that is a bijection. My goal is to prove the following, Suppose $A \subseteq ...
0
votes
1answer
25 views

Prove that $\prec$ is irreflexive and transitive

Note: Definitions I use (Velleman's How To Prove It) If $A$ and $B$ are sets, then we will say that $B$ dominates $A$, and write $A \precsim B$, if there is a function $f: A \rightarrow B$ ...
0
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1answer
38 views

Are $x$ and $y$ divisible by $n$, if so how do I prove it?

If $y$, $x$,are natural numbers, and $n$ is a prime number, $y = x + n$, $y>x>n$, and $y$ and $x$ are not coprime, is it true that $n$ is a divisor of both $x$ and $y$? If so could you please ...
11
votes
1answer
164 views

A proof involving the Euler phi function

Problem: Let $\varphi$ be the Euler phi function, where for any $n \in \mathbb{Z^+}$, $\varphi(n)$ is the number of positive integers less than $n$ that are relatively prime with $n$. ...
-2
votes
0answers
24 views

To prove ($A\cup B$) $\cap C$ = $(A \cup C) \cap (B \cup C)$ [duplicate]

I have never done rigorous et theory before .How do i prove this and generalise for $A_{i}$ ,i belonging to I ($A\cup B$) $\cap C$ = $(A \cup C) \cap (B \cup C)$ Hints ? Thanks
1
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1answer
59 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
0
votes
3answers
29 views

Proving $T(n) = 1 + \sum_{j=0}^{n-1} T(j)$, $T(0)=1$ implies $T(n)=2^n$

I feel that this is a fundamental question. $$ T(n) = 1 + \sum_{j=0}^{n-1} T(j). $$ Given $$ T(0) = 1. $$ Show $$ T(n) = 2^n. $$ If I substitute values, I can see that the series goes like 1, 2, 4, ...
0
votes
1answer
30 views

What sequence of polynomials is equal to $2^n$ for integers $1$ to $k$?

I am trying to prove to someone that no matter how many terms you have of a sequence you can never be 100% sure of the underlying formula. Consider this sequence: $$2^n=1,2,4,8,16,...$$ But just given ...
4
votes
2answers
62 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
3
votes
2answers
56 views

Help in proving a tautology

I am having real trouble deriving this tautology: $\forall(x) ((x=a) \lor (x\neq a))$ It is easy to solve this by assuming the negation, unpack the negation with DeMorgan's Law, and derive from ...
0
votes
1answer
53 views

Proof of infinite monkey theorem. [duplicate]

I was just wondering, does the infinte monkey theorem also has a proof? And why is this called a theorem? It is sheer common sense. And what are its applications. I have heard about PHP and IEP and I ...
0
votes
2answers
50 views

Construction of Natural Numbers

I am trying to prove that the natural numbers can be constructed from the product of a power of $2$ and an odd number. For all $n \neq 0$ in the natural numbers, $n = (2k+1)(2^p)$, where $k$ and $p$ ...
2
votes
1answer
42 views

Do all singular $n\times n$ matrices form a vector subspace when $n\ge2$?

I want to prove or disprove that the set of all $n\times n$ singular matrices form a vector subspace of $M_{nn}$ when $n\geq 2$. So, let: $$ A_{n,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & ...
0
votes
1answer
23 views

Measurability of sequence of functions

Let $(f_n)_{n \in \Bbb N}$ be a sequence of measurable functions on a measure space $(X, M, \mu)$. Prove that the set $\{x \in X \; | \; \lim_n f_n(x) \text{ exists} \text{in } [-\infty, ...
0
votes
1answer
35 views

Prove that $\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$

Let $a,b,c$ be complex numbers such that $|a+b|=m$ and $|a-b|=n$ and $mn\ne0$. Prove that $$\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$$ I have tried using formula ...
12
votes
4answers
193 views

Proving $\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}} < 3$ for $n\geq 1$ by induction

How can I prove by induction that $\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}} < 3$ for $n\geq 1$? My guess is that there must be another form to express the sum of nested square roots, but I don't know how ...
-4
votes
2answers
52 views

Proof By Induction [on hold]

I am trying to prove the Following, However, I dont understand what to do at the Inductive Step: Any Help would be appreciated!
4
votes
2answers
106 views

How do I prove that $ f(n) = (n + 1)! - 1 $ is an injective function?

I have this problem: Consider the function $f : \mathbb{N} \rightarrow \mathbb{N}$ defined, for every $n \in \mathbb{N}$, by $$f(n) = (n+1)! - 1$$ Prove that $f$ is injective. How do I ...
1
vote
2answers
23 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
1
vote
1answer
48 views

Prove that $f$ is an onto function and a homomorphism function from $(\mathbb{Z} \times \mathbb{Z}, \oplus)$ to $(\mathbb{Z}, +)$

I have a lot of issues trying to figure out this problem. Any advice? Consider the two groups $(\mathbb{Z} \times \mathbb{Z}, \oplus)$ and $(\mathbb{Z}, +)$, where $(a,b) \oplus (c,d) = (a + c, b + ...
3
votes
0answers
34 views

Proof that $\mathbb{Z}[\sqrt{3}]$ is a Euclidean Domain.

Proof that $\mathbb{Z}[\sqrt{3}]$ is a Euclidean Domain. Proof that $\mathbb Z[\sqrt{3}]$ is a Euclidean Domain Is it possible to solve this question without using $\mathbb{Q}[\sqrt{3}]$ restricted ...
1
vote
4answers
52 views

Proof that $0.33333… = \frac{1}{3}$ using $\epsilon-N$ method

This proof is quite prevalent on the web, yet I struggle using this particular method. Wikipedia (http://en.wikipedia.org/wiki/Limit_of_a_sequence) tells us: We call $x$ the limit of the sequence ...
1
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0answers
42 views

Is this simplification 'allowed'?

I've just been doing a problem that involved this equation: $$ \frac{1}{\sin\left(\frac{\theta}{2}\right)}\left( \sin\left(b\theta-\frac{\theta}{2}\right)-\sin\left(a\theta-\frac{\theta}{2}\right) ...
2
votes
1answer
16 views

Proving $\alpha\colon S\to T$ is one-to-one if $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$, where $A,B\subseteq S$

Prove that $\alpha\colon S\to T$ is one-to-one if $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$. Book solution: Assume that $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$ for every pair of subsets $A$ and ...
2
votes
1answer
31 views

Proving that if $S$ has an infinite subset then $S$ is infinite

Definition$\quad$ A set $S$ can be defined as infinite if there exists a mapping from $S$ to $S$ that is one-to-one but not onto. Otherwise, $S$ is finite. Problem: Using the definition of ...
0
votes
1answer
26 views

Nim Variant - Strong Induction Proof

Here we will play a variant of Nim where there is an additional move option in some cases. If two or more piles have the same number of stones, a player may remove the same number of stones from ...
9
votes
2answers
289 views

Lebesgue integration of simple functions

Define $f : [0,1] \to \Bbb R$ by $f(x) := 0$ if $x$ is rational, and $f(x) := d^2$ if $x$ is irrational, where $d$ is the first nonzero digit in the decimal expansion of $x$. Show that ...
-1
votes
3answers
45 views

Summation of $3^k$ from $2$ to $72$ [on hold]

i'm currently stuck on the following question and am not supposed to be using a summation calculator to find the answer: $$ \sum _{k=2}^{72}\left(3^k\right) $$ Please could somebody explain to me ...
0
votes
1answer
29 views

$n$ divides $a_1 - a_2$ as well as $b_1 - b_2$. Show that $n$ divides $a_1b_1 - a_2b_2$.

I keep arriving at $a_1b_1$ and $a_2b_2$ having the same sign if I multiply the equations $a_1 - a_2 = nk$ and $b_1 - b_2= np$ times each other. They must be opposite signs so that I can say that $n$ ...
0
votes
1answer
16 views

Proving results using Fundamental theorem of calculus

If $g(x) = x$ for $\lvert x \rvert \ge 1$ and $g(x) = -x$ for $\lvert x \rvert < 1$ and if $G (x) = \frac{\lvert x^2-1 \rvert}{2} $, show that $$\int^3_{-2} g(x) dx = G(3) - G(-2) = ...
1
vote
2answers
29 views

Which of the properties, Reflexive, Irreflexive, Symmetric, Asymmetric, Antisymmetric, Transitive, Linear, does F satisfy?

Let $S={(n,m) ∶n,m∈Z^+}$. Define the relation F on S by ${(n,m),(i,j)}∈F$ if and only if $nj=mi$. In other words, let $F = {((n, m), (i, j)) ∈ S × S: nj = mi}$. Proof F is reflexive: Show that for ...
0
votes
0answers
14 views

Free harmonic vibrations of the Euler-Bernoulli equation

The Euler-Bernoulli equation describes the relation between external forces and deflections of a beam. The general formula is given by: $$ \frac {\partial ^2}{\partial x^2} \left(EI\frac{\partial ...
0
votes
0answers
7 views

Riemann sum and partitions

If f is riemann integrable and if $(P_n)$ is any sequence of tagged partitions of [a,b] such that $\lVert P_n \rVert$ -> 0, prove that $\int_a^b f = lim_n S (f;P_n)$. I am confused as to how to ...
1
vote
2answers
23 views

Show an absolute minimum and positive/negative derivative of function

Let $f : \mathbb R \to \mathbb R$ be defined by $f(x) := 2x^4+x^4\sin(1/x)$ for $x \neq 0$ and $f(0) = 0$. Show that f has an absolute minimum at x = 0, but that its derivative has both positive and ...
1
vote
1answer
22 views

Using mean value theorem for multiple inequalities

Use the Mean Value Theorem to prove that $\frac{(x-1)}{x} < \ln x < x-1$ for $x > 1$. I was thinking of breaking up the inequality into \frac{(x-1)}{x} < \ln x$, and $\ln x < x-1$ and ...
3
votes
3answers
57 views

Series Proof $\sum_{k=1}^n (1/k) > \ln(n+1)$

Prove that $\sum_{k=1}^n (1/k) > \ln(n+1)$. I have been trying to do this for some time now, but I cannot figure it out. It is on the study guide for my final exam, which is tomorrow so I am trying ...
-2
votes
0answers
23 views

Intro to Number Theory, Simple Continuous Fractions Question? [on hold]

I have no idea how to start this question, any help would be appreciated! Show that $k_n|k_{n-1}\alpha-h_{n-1}|+k_{n-1}|k_n\alpha-h_n|=1$
2
votes
3answers
35 views

Determine where h(x) := x $\lvert x \rvert$ is differentiable from R to R.

Determine where h(x) := x $\lvert x \rvert$ is differentiable from R to R. Not totally sure how to start this. Much appreciation, Jesse
-3
votes
0answers
27 views

Differentiable Function at c =0? [on hold]

Suppose that $f : \mathbb{R} \to \mathbb{R}$ is differentiable at $c$ and that $f (c) = 0$. Show that $g (x) := \lvert f(x) \rvert$ is differentiable at $c$ if and only if $f'(c) = 0$.