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
votes
3answers
17 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, ...
0
votes
1answer
28 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
59 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
51 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
35 views

Proof of infinite monkey theorem.

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
0answers
36 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
41 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
21 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
33 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
187 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
50 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
39 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
51 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
vote
0answers
40 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
285 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
39 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
28 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
56 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
21 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 $f'(c) = 0$.
0
votes
1answer
24 views

Finding an Injection

I need to prove the set A={1/n: n$\in$$\mathbb{Z}\backslash${0}} is countably infinite. To prove it is infinite, I said consider the set B={1/n: n$\in$$\mathbb{Z}^+$}, and note that B$\subseteq$A. ...
-5
votes
3answers
41 views

If the limit of the sequence exists, find it. If not, prove that the limit does not exist. [on hold]

Consider the following sequence: $\{[\sqrt{n}][\sqrt{n + 1}-\sqrt{n}]\}$ for $ n \geq 1$. If the limit exists, find it and prove that the limit is indeed your choice. If not, prove that the limit ...
0
votes
2answers
33 views

Show that if $f$ and $g$ are uniformly continuous on $A\subseteq\mathbb{R}$, then $f + g$ is uniformly continuous on $A$. [duplicate]

Show that if $f$ and $g$ are uniformly continuous on $A \subseteq\mathbb{R}$, then $f + g$ is uniformly continuous on $A$. How do I approach this question?
0
votes
2answers
27 views

Weird continuity proof

Let $I = [a,b]$ and let $f : I \to \Bbb R$ be a continuous function on $I$ such that for each $x$ in $I$ there exists $y$ in $I$ such that $| f(y)|\le | f(x)|/2$. Prove there exists a point $c$ in ...
0
votes
3answers
54 views

Prove that $A$ is countable.

Hi so I'm practicing for a exam and I need help to figure this proof out, Suppose $A\subseteq \mathbb R^+$, $b\in\mathbb R^+$, and for every list $a_1,a_2,\ldots,a_n$ of finitely many distinct ...
2
votes
1answer
32 views

Proving a set to be countably infinite.

I'm asked to decide and prove whether the set $\{\,x\in\mathbb{N}: |x-7|>|x|\,\}$ is finite, infinitely countable, or uncountable. I'm pretty certain it is infinitely countable. I say that since ...
13
votes
3answers
150 views
+200

How to prove that it is possible to make rhombuses with any number of interior points?

I was given some square dot paper which can be found on this link: http://lrt.ednet.ns.ca/PD/BLM/pdf_files/dot_paper/sq_dot_1cm.pdf and was told to draw a few rhombuses with the vertices on the dots ...
1
vote
1answer
35 views

Riemann Sum proofs

If $f$ is Riemann integrable on $[a,b]$ and $\lvert f(x) \rvert$ $\le$ $M$ for all $x$ $\epsilon [a,b]$, show that: $\lvert \int_a^b f \rvert$ $\le$ $M(b-a)$ Just started learning Riemann sums ...
2
votes
2answers
38 views

Derivative Definition proofs

Let $f : \Bbb R \to \Bbb R$ be defined by $$f(x)=\begin{cases}x^2, & \text{if $x$ is rational} \\ 0, & \text{if $x$ is irrational} \end{cases}$$ Show that $f$ is differentiable at $x = 0$ and ...
0
votes
1answer
35 views

A proof about boundedness for continuous functions

Let $I := [a,b]$ and let $f : I \rightarrow \mathbb{R}$ be a continuous function such that $f(x) > 0$ for each $x$ in $I$. Prove that there exists a number $a > 0$ such that $f(x) \geq a$ for ...
0
votes
1answer
15 views

Boundedness Theorem for continuous functions on intervals

Just want to confirm this is a suitable proof: Assume $f$ is not bounded on $I$. So, for any $n \in \mathbb{N}$, $\lvert f(x)\rvert > n$. Since $I$ is bounded, $x_n$ is also bounded. By ...
-4
votes
4answers
27 views

Let $f,g$ be continuous from $\mathbb R$ to $\mathbb R$ [duplicate]

Let $f, g$ be continuous from $\mathbb R$ to $\mathbb R$, and suppose that $f(r) = g(r)$ for all rational numbers $r$. Is it true that $f(x) = g(x)$ for all $x \in \mathbb R$?
3
votes
2answers
341 views

Determine if the following is surjective

I need to determine if $f: \Bbb N\times\Bbb N \to \Bbb N$ such that $f(a,b) = a^b$ is a surjective (onto) function. My intuition is that it is but I don't know how to prove it. I don't even know how ...
2
votes
2answers
36 views

Prove that Set B is countable - Is this proof correct?

It seems that I have some issues with the rigor of this proof and I don't know what I'm doing wrong. Could someone tell me if this proof is correct and rigorous enough? Here's the question Prove ...
1
vote
2answers
28 views

Given a complete graph of n vertices Kn (has all possible edges – one edge between pair of vertices).

Given a complete graph of n vertices $K_n$ (has all possible edges – one edge between pair of vertices). Use counting to find a formula in $n$ for the number of edges in the graph. I know that the ...
1
vote
2answers
21 views

Cardinality of the union of two sets

I am having trouble attempting to prove the inequality $|X\cup Y| \le |X|+|Y|$. Here is my intuitive argument when we take the union of $X\cup Y$ if there are repeated elements then they are not ...
1
vote
2answers
43 views

How to prove countably infinite?

How do I prove the following set is countably infinite? $\{\frac{1}{n}: n\in\mathbb{Z}\setminus\{0\}\}$ I know that I can say this set is a subset of $\mathbb{Q}$, and that $\mathbb{Q}$ is infinite, ...