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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0
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2answers
72 views

Solutions of $\arctan x = 1 - x$. Proofs?

In this question, we examine the equation $\arctan x = 1 - x$. You may assume without proof that $\arctan x$ is continuous on $R$. a) Prove that there is a solution to the equation in the interval ...
0
votes
1answer
63 views

How to prove this form of $n$?

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. From: AOPS Putnam A1 Solution I ...
0
votes
3answers
249 views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
0
votes
3answers
113 views

Derive Closed form sum of N^2

Can anyone explain to me how you would derive this ? I have this question asked in a CS class and can't figure out how to derive it. it has to be derived as you would with sum of N ex ...
0
votes
1answer
281 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
12
votes
3answers
2k views

Show that the product of two consecutive natural numbers is never a square.

I'd like to have my proof verified and if possible, to see other solutions that are interesting. Proof: Suppose $n(n+1)$ is a square. Then we write $$n(n+1) = \prod_{p} p^{c(p)}$$ where $c(p) = a(p) ...
8
votes
1answer
834 views

Proof of a theorem of Cauchy's on the convergence of an infinite product

Well it is relatively well known that the condition for absolute convergence is given by the following theorem: In order that the infinite product $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ may be ...
7
votes
3answers
249 views

How to understand proof of a limit of a function?

Given the following function: $$ f(x)=\left\{ \begin{array} {cc} 0, & x \text{ irrational, } 0<x<1 \\ \frac{1}{q}, & x=\frac{p}{q} \text{ in lowest terms, } 0<x<1 \end{array} ...
7
votes
5answers
897 views

Constructive proof of boundedness of continuous functions

Consider the theorem for the continuous function: Let $a<b$ be real numbers, and let $f:[a,b]\to{\bf R}$ be a function continuous on $[a,b]$. Then $f$ is a bounded function. The proof in the ...
6
votes
3answers
110 views

How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction

I've done the base step, but how do I prove it is true for $n+1$ without using a fallacy? $$4(n!)>2^{n+2}\quad \text{for } n\geq 4$$ Please help.
5
votes
4answers
496 views

Discrete Math Proofs Involving Real Numbers

I am stuck on these two problems. $1$. Prove that for every three positive real numbers a, b, and c that $(a+b+c)*(\frac{1}{a}+\frac{1}{b} + \frac{1}{c}) \ge 9$. $2$. Prove that for every three ...
5
votes
3answers
285 views

Help with Cartesian product subsets [duplicate]

I want to prove that if $A \subseteq C\,$ and $\,B \subseteq D,\,$ then $\,A \times B \subseteq C \times D.$ I know that $A \subseteq C \iff a \in A \rightarrow a \in C$ and that $B\subseteq D\iff ...
5
votes
4answers
293 views

Combinatorial Proof

I have trouble coming up with combinatorial proofs. How would you justify this equality? $$ n\binom {n-1}{k-1} = k \binom nk $$
4
votes
3answers
227 views

Invertibility in a finite-dimensional inner product space

Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{\star}$ is also invertible and $( T^{-1} )^{\star} = ( ...
4
votes
3answers
5k views

Prove that the intersection of two equivalence relations is an equivalence relation.

I am reading this chapter of the Book of Proof, and I'm stuck at the Exercise 10 of section 11.2. It is as follows. Suppose $R$ and $S$ are two equivalence relations on a set $A$. Prove that $R ...
3
votes
1answer
97 views

The number of regions into which a plane is divided by n lines in generic position [duplicate]

Suppose that $n$ lines are drawn on a plane in such a way that no lines are parallel and no three of them intersect at a point. Let $r(n)$ be the number of regions the plane is divided into after ...
3
votes
1answer
52 views

Do absolute convergence of $a_n$ implies convergence of $K_n=\frac{1}{\ln(n^2+1)}\sum_{k=1}^{+\infty}a_k\frac{3k^3-2k}{7-k^3}\sin k$?

The problem asks us to decide whether the following statement is true: Let $\{a_n\}_{n\geq1}$ be any absolutely convergent sequence. Does that imply that the sequence: ...
3
votes
3answers
76 views

Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism

I'm working on the following problem: Let $G$ be a group. Prove that $G$ is abelian if and only if $\varphi(g) = g^2$ is a homomorphism. My solution: First assume that $G$ is an abelian group ...
3
votes
2answers
927 views

How to find the period of the sum of two trigonometric functions

I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$
3
votes
2answers
579 views

Why is $\cos(2x)=\cos^2(x)-\sin^2(x)$ and $\sin(2x)=2\sin(x)\cos(x)$?

I was studying math.. and I just realized that I only just memorized these trigonometric equations, but I don't really know the reason behind them. So um... Why is $\cos(2x)=\cos^2(x)-\sin^2(x)$ and ...
3
votes
2answers
137 views

Let $S$ and $T$ be finite non-empty sets such that $|S| = |T|$. Show that the function $f : S\to T$ is onto if and only if it is one-to-one.

This is a recent homework bonus question assigned in my Proofs and Conjectures class. It (evidently) includes and evaluates our understanding of elementary-set theory and how to determine and prove ...
3
votes
2answers
135 views

If $f$ is differentiable and $\lim_{x→0} f'(x) = L$, then $f'(0) = L$.

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x→0} f'(x) = L$, then $f'(0) = L$. 1. How to presage proof by contradiction? Proof by contradiction. ...
3
votes
2answers
108 views

Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G]. $ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
3
votes
4answers
106 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
3
votes
1answer
122 views

Proof for recursively defined sets

Language $L\subset \{a,b\}^*$ is such that: $\epsilon \in L$ $a \in L$ For any $x\in L$, $xb\in L$ and $xba\in L$ Nothing else in $L$. Im just learning recursive sets, but with that definition am ...
3
votes
2answers
127 views

Prove through induction that $3^n > n^3$ for $n \geq 4$

I'm new to induction and have not done induction with inequalities before, so I get stuck at proving after the 3rd step. The question is: Use induction to show that $3^n > n^3$ for $n \geq ...
3
votes
2answers
343 views

Prove the monotonicity of the expectation of a binary random variable function

Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
3
votes
2answers
2k views

Prove the centralizer of an element in group $G$ is a subgroup of $G$

We have a group $G$ where $a$ is an element of $G$. Then we have a set $Z(a) = \{g\in G : ga = ag\}$ called the centralizer of $a$. If I have an $x\in Z(a)$, how do I go about proving that the inverse ...
2
votes
2answers
73 views

Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7 [closed]

I tried proving this by contraposition, by saying, "If every number that is being averaged is greater than 7, then the average of a thousand numbers is less than 7." This seems easier to prove, but I ...
2
votes
2answers
550 views

If $x \in\mathbb{Z}$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$

I am learning proofs, and I am stuck with this proposition: Let $x \in\mathbb{Z}$. If $x$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$. I want to use the additive ...
2
votes
2answers
103 views

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction [duplicate]

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction My proof so far: Let $P(n)$ be $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ Base Case $P(1):$ LHS = $1^3 = 1$ ...
2
votes
2answers
41 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
2
votes
7answers
125 views

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z.$ I know this is true because any even number that is squared will be even, is it also true than any even number ...
2
votes
2answers
153 views

Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$

I think I have the first part, as I have shown that $q < q + (q'-q)/2 < q'$, but I have trouble in proving that $|q-q'|< \epsilon$. Could someone tell me if there is a better way of showing ...
2
votes
1answer
253 views

Pigeonhole Principle

Let $X = {x_0, x_1, · · · , x_m}$ be a subset of ${1, 2, · · · , n}$, where $m > n/2$, and $x_0$ is the smallest number in $X$. Use the pigeonhole principle to show that $X$ contains two numbers ...
2
votes
1answer
116 views

Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
2
votes
2answers
312 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
2
votes
2answers
226 views

$\binom{n}{k}\binom{n}{n - k}$ vs $\binom{n}{k}$ - Differences? Similarities?

So the # of ways to choose an $n$ set with $k$ kiwis is $\binom{n}{k}\binom{n}{n - k} = \binom{n}{k}^2$. AlexR wrote No, picking exactly $k$ kiwis means you discount the $n-k$ remaining ...
2
votes
1answer
25 views

Reduce all cases to $x \to 0^{+}$ and $f(x),g(x) \to 0$ before proving L'Hôpital's Rule

Hypotheses: I. $f$ and g differentiable on $(a,b)$ and continuous on $(a,b]$, II. $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0 \quad $ or $\pm \infty$ III. $\lim_{x \to a} f'(x)/g'(x)$ exists IV. ...
2
votes
0answers
238 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
2
votes
3answers
330 views

Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$

I haven't been able to do this exercise: Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X ...
2
votes
1answer
85 views

Proofs involving sequential limit

Let $S$ be the domain of the function $f$. Suppose $S=\left\{\frac{1}{n}: n\in\Bbb N\right\}$. Show $\lim_{x\to0}f(x)=L$ iff $\lim_{n\to\infty}f\left(\frac{1}{n}\right)=L$. Idea: I want to say, let ...
2
votes
2answers
125 views

Proof that this is independent

Prove that {$1, \sin(x), \sin(2x), \sin(3x),\ldots, \sin(nx)$} is an independent set. I can think of the long way which is to differentiate this and put the differentiations into a matrix and row ...
2
votes
1answer
122 views

Where is the fallacy in this coupling argument of two Bernoulli variables?

With respect to the scenario introduced in Prove the monotonicity of the expectation of a binary random variable function, let us now suppose that the function: $$\begin{align*} f(\mathcal{J}) = ...
2
votes
3answers
266 views

I don't understand this proof of the AM-GM inequality?

The proof uses this lemma which I understand: $\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ ...
2
votes
1answer
941 views

lim sup inequality proof - is this the right way to think?

I have tried to read many proofs of this but I'm not sure I get it, so please bare with me. Show that $\lim_{n \rightarrow \infty} \sup (a_n+b_n) \leq \lim_{n \rightarrow \infty} \sup (a_n)+lim_{n ...
2
votes
2answers
97 views

L shaped region

Prove, using the well-ordering principle, that, for all $n\geq 1$, an $\mathsf{L}$-shaped space with two sides of length $2n$ and four sides of length $n$ can be tiled using some number of 3 square ...
2
votes
4answers
138 views

Is a contradiction enough to prove a set equality to $\varnothing$?

An exercise asks me to prove the following: $$A \cap (B-A) = \varnothing$$ This is what I did: I need to prove that $A \cap (B-A) \subseteq \varnothing$ and $\varnothing \subseteq A \cap (B-A)$ The ...
2
votes
2answers
145 views

Sum of Sines Interval [duplicate]

Possible Duplicate: How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? How is it possible to show for integer $m$: ...
1
vote
4answers
53 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 ...