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.

learn more… | top users | synonyms

0
votes
0answers
8 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
3
votes
2answers
23 views

How do I prove that $\forall \beta\in F(\alpha)\setminus F$ is transcendental?

Let $E/F$ be a field extension. Let $\alpha\in E$ be transcendental over $F$. Let $\beta\in F(\alpha)\setminus F$. Then, how do I prove that $\beta$ is transcendental over $F$? Here's how I tried: ...
0
votes
0answers
12 views

Prove multivariable function is injective?

I am a little confused on how to prove a multivariable function is injective(one to one). I know the process for single variables but got stuck sadly. The function f: N -> N such that f((a,b)) = a^b ...
2
votes
4answers
364 views

How do I write this proof formally?

How can I formally prove that $$\max\{\lvert x+y\rvert _i \} \leq \max\{ \lvert x_j \rvert \} + \max\{\lvert y_k \rvert \}$$ Where $x,y$ are the components of a $n$-vector with $1 \leq i,j,k \leq ...
1
vote
1answer
23 views

Dijkstra's Algorithm for Negative Weights.

Now the problem states that their is a graph $ G = (V,E) $ where some of the edges have negative weights while some of the edges have positive edges. Now the question is why won't Dijkstra's algorithm ...
5
votes
2answers
39 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
0
votes
0answers
18 views

Argument for finite solution of power Diophantione Equation.

Assume the equation $4x^3=y^2+3$ has infinite positive integer solution. If $x,y$ has general solution then it is clear that for any $x$(rational, integer), there is a $y$. It can be said there is a ...
0
votes
1answer
19 views

Cholesky factorization exist?

Is there a theorem or a way to show that if I have a real and symmetric positive definite matrix $A$ and its Cholesky factorization is $A = LL^T$ then $B = L^TL$ is also positive definite? Or in other ...
0
votes
2answers
35 views

Use the Mean Value Theorem to prove [on hold]

Use the Mean Value Theorem to prove that $|\sin x - \sin y| \leq |x - y|$ for all $x,y \in \mathbb{R}$.
1
vote
1answer
25 views

Prove that set of isolated points in $X$ is dense in $X$

Let $A=\{\text{isolated points of } X\}$. $X$ is a countable complete metric space. Show that $A$ is dense in $X$. My attempt: Basically we want to show that $\bar A = X$. First, we show that ...
0
votes
2answers
33 views

Power set equinumerosity. Is this proof correct?

So I'm trying to prove the following, Prove that if $A\sim B$ then $\mathscr{P}(A) \sim \mathscr{P}(B)$. Here's how I started out to prove there is a function that is injective: Suppose $A ...
3
votes
4answers
53 views

Proving $2^n -1 = \sum_{i=0} ^{n-1} 2^i$ for all $n\geq 1$ by induction

I'm practicing proofs by induction, and equalities seem to be the toughest for me. Can somebody please help to prove that for all integers $n \geq 1$: $$ 2^n -1 = \sum \limits _{i=0} ^{n-1} 2^i\;? $$ ...
2
votes
1answer
17 views

An interval with width greater than one contains an integer.

If I have an interval $(a, b)$ such that $b - a > 1$, how can I prove that this contains an integer? It seems 'obvious', but a formal proof eludes me.
4
votes
5answers
96 views

Proving $6^n - 1$ is always divisible by $5$ by induction

I'm trying to prove the following, but can't seem to understand it. Can somebody help? Prove $6^n - 1$ is always divisible by $5$ for $n \geq 1$. What I've done: Base Case: $n = 1$: $6^1 - 1 = ...
0
votes
4answers
64 views

Proof by Induction $3^n > n^3$

I am trying to prove the following, however I'm stuck at the Induction hypothesis Prove by induction that, for all integers $n$, if $n\geq 5$, then $3^n>n^3$ What I have Done: Base Case: $n ...
11
votes
2answers
689 views

Inverted induction

I am working on a proof, and to do it, I think it would be optimally to use induction backwards. Show that 1 doesn't work. Assume n doesn't work. Proof that n+1 doesn't work. Is this valid?
-1
votes
1answer
28 views

Prove that $2n + m \equiv 0 \pmod3$if and only if $ n \equiv m \pmod3$ [on hold]

Prove that $2n + m \equiv$ $0 \pmod3$ iff $n \equiv$ $m \pmod3$ Is there a way to prove this without proving both directions of the biconditional?
2
votes
1answer
37 views

Proving $k$ is divisible by $3$ iff the sum of the digits of $k$ is divisible by 3 [duplicate]

I am trying to prove that $k$ is divisible by $3$ iff the sum of the digits of $k$ is divisible by 3 for all $k \in Z$. I am not even sure what tags to use because I am not sure of right methods to ...
1
vote
2answers
41 views

Summation Proof Dealing With 3s Multiples [duplicate]

So the problem is as follows: Prove that if the sum of digits of a decimal $n$ is three's multiple, then n is three's multiple by direct proof. For example, $11234567$ is 3's multiple because ...
-2
votes
0answers
47 views

Summation Direct Proof Help [on hold]

Prove that if the sum of digits of a decimal n is three's multiple, then n is three's multiple by direct proof. For example, 11234567 is 3's multiple because 1+1+2+3+4+5+6+7=24, and in fact, 11234567 ...
0
votes
1answer
33 views

How to prove this theorem? (Logical symbols help)

This is a theorem from a book. I'm having a hard time on proving it. Suppose A is a set,$\mathcal{F}\subseteq \mathscr{P}(A)$, and $\mathcal{F} \neq \emptyset$. Then the least upper bound of ...
0
votes
2answers
40 views

Dot and Cross Product Proof: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$

How do you prove that: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ ? The textbook says as a hint to "first do it for $u=i,j$ and $k$; then write $u-xi+yj+zk$ but I am not sure what that ...
1
vote
2answers
36 views

Proving $10^n \equiv 1 \pmod 3$ for all $n\geq 1$ by induction

Prove that $10^n \equiv 1 \pmod 3$ for all positive integers $n$ by mathematical induction. Can someone please help me in solving this problem and explain what's going on? Any guidance would be ...
4
votes
4answers
123 views

How to prove that the value of $e$ is irrational without using the number $e$ itself [on hold]

Recently I have tried to prove that the value of $e$ is irrational without using the number $e$ itself. I have seen that the number $e$ can be expressed as $$\lim_{n\to\infty}(1 + 1/n)^n;$$ however, ...
1
vote
2answers
60 views

Proving $A$ is a subset of $B$

I'm trying to understand the proof behind showing a set is a subset of another set, but I'm struggle to do so. Can some one help using this example to show: $A \subseteq B$? Here $A = \{x | x = 4n ...
0
votes
3answers
47 views

Integral equal to Riemann Zeta Function

As part of a homework problem in Rudin, I need calculate $$ \int_{1}^{N} \frac{[x]}{x^{s+1}} \,dx$$ where $[x]$ is the floor function. Clearly $[x]$ has derivative $0$ everywhere but the integers. ...
-2
votes
1answer
30 views

Combinations of continuous functions

Determine the points of continuity: $g(x) := \sqrt{x+\sqrt{x}}$, where $x\ge 0$. I am having trouble even defining the composite functions to begin with... *Brain Freeze
0
votes
1answer
29 views

Continuity proof with epsilon-delta

Let $K > 0$ and let $f:\ \mathbb R\to\mathbb R$ satisfy the condition $\lvert f(x) - f(y)\rvert \leq K \lvert x - y\rvert\ \forall x, y \in \mathbb R$. Show that $f$ is continuous at every point ...
1
vote
4answers
88 views

If $a$ and $b$ are positive integers and $4ab-1 \mid 4a^2-1$ then $ a=b$.

Prove that if $a$ and $b$ are positive integers and $$(4ab-1) \mid (4a^2-1)$$ then $a=b$. I am stuck with question, no idea. Is there any way to prove this using Polynomial Division Algorithm? Would ...
0
votes
1answer
13 views

Using existential instantiation on a universally quantified given

I'm trying to prove the following exercise of How to Prove it: A structured Approach (Section 3.4, exercise 19): Suppose A, B and C are sets. Prove that A $\triangle$ B and C are disjoint iff A ...
0
votes
4answers
118 views

Is $ (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D)$ true for all sets $A, B, C$ and $D$?

Is $(A \times B) \cup (C \times D) = (A \cup C) \times (B \cup D)$ true for all sets $A, B, C$ and $ D?$ I tried to wrap my head around this, but I have absolutely no idea what is going on here. How ...
1
vote
1answer
71 views

Two definitions of topological entropy: Why do they coincide?

I guess you all know the definition of topological entropy by using open covers for $X$ being a compact topological space and $T\colon X\to X$ being a continuous map (for example given in Walters' "An ...
0
votes
3answers
30 views

Is the following set operation true?

Prove the following or else find a counter example: For all sets $A$, $B$, and $C$, $$((A \cup B) − C) \cup (A \cap B) = ((A − B) \cup (B − A)) − C$$ For the life of me, I can't figure out if its ...
-2
votes
0answers
22 views

Proving finite/infinite sets

For j$\in\mathbb{Z}^+$, let $A_j$$\subseteq$$\{$1,..., j$\}$. Suppose that for some n$\in$$\mathbb{Z}^+$, we have B$\subseteq$$\cup^{1}_{j=1}$$A_j$. Is B necessarily finite? Prove it or give a ...
0
votes
2answers
79 views

Proving a set is infinite [on hold]

Prove that $\mathbb{Q}$ is infinite. I think the strategy is to assume it is finite, and then prove by contradiction?
1
vote
2answers
51 views

If $x,y \in \mathbb{R}$ where $x\leq y$ and $y\leq x$. Does $x=y$?

I'm trying to complete this problem: Let $A$ be a nonempty set and suppose $\alpha$ and $\beta$ are both suprema of $A$. Prove that $\alpha = \beta$. The first thing i did was try to find an ...
0
votes
0answers
12 views

Upper bounds and Lower bounds (Relations Proof Problem)

So I've only recently started studying proofs and I've been using Velleman's "How to Prove it" This is a theorem from the book. I'm having a hard time on proving it. Suppose A is a ...
0
votes
2answers
32 views

Limit of monotonic function

I have to prove that if $(x_1 \gt x_2) \Rightarrow (f(x_1) \ge f(x_2))$, then $$\forall a \in \mathbb R \exists L \in \mathbb R \lim_{x \to a^+}f(x) = L$$ I have a feeling that L = $inf_{x \in (a,a+ ...
1
vote
2answers
40 views

Non-existing Limit of $\sin x$

How do I prove from definition of limit that $\lim_{x \to \infty}\sin x$ is non-existant? I tried to negate said definition: $$\lnot ((\exists L)(\forall\epsilon)(\exists \delta):(\forall x)(|x|\gt ...
0
votes
0answers
27 views

Applying chain rule in probability?

Let $X,Y$ be random variables with distribution functions $F_X(x)$, $F_Y(y)$. Let $W(u,v)=max\{0,u+v-1\}$. why can we take the following limits "inside" $W$? $lim_{(x,y)\to ...
2
votes
1answer
38 views

If $\int_t ^{t+1}f=1$, then showing $f(x+1)=f(x)$

Let $f$ be locally integrable on the real line and $\int_t ^{t+1}f$ is contstant for $t\in \mathbb{R}$, then $f\left(x+1\right)=f\left(x\right)$ almost everywhere. I don't see how Lebesgue's ...
0
votes
2answers
51 views

Prove or disprove: There exists a prime p > 3 such that p + 2 and p + 4 are also prime

I'm having a lot of difficulties with this proof. Can someone please solve it and explain to me what's going on at each step? Thank you!
2
votes
2answers
246 views

how to prove that the following is not a regular language?

the language we want to disprove is : $$ L = \{ 0^i1^j| gcd(i,j)=1 \} $$ my attempt : i used the pumping lemma this way: consider the set of strings of the form $0^p1^q$ such that $n <=p$ and ...
4
votes
2answers
93 views

How much does Proof writing improve over the years?

This is a very soft question. Just a bit of background: I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his ...
1
vote
0answers
22 views

Let A= [0,1] - {1/n │n ∈ N}. Find sup(A), inf(A), min(A), max(A).

My idea of this question is to claim sup(A) and inf(A) exists (and equals a value) and prove by contradiction that min(A),max(A) exists afterwards (and equals sup(A),inf(A)). The issue that I have is ...
2
votes
2answers
72 views

Prove $f(x)=g(x)$ for all $x \in\mathbb{R}$

If $$f(x)=\sum_{n=0}^\infty\frac{x^n}{n!}, x\in\mathbb{R}$$ and $$ g(x) = 1 + \int_0^x f(t) \,dt $$ prove that $g(x)=f(x)$ for all $x\in\mathbb{R}$ and prove that $f$ is differentiable on ...
0
votes
1answer
67 views

How to prove that the matrix $A^k$ approaches $0$ as $k$ approaches infinity

First of all, what does it mean to say an eigenvalue is "less than unity"? I'm not exactly sure what this means. Secondly, how do I show that $\lim_{k\to\infty} A^k=0$ given that all eigenvalues of ...
0
votes
1answer
32 views

Prove that if $\lim_{x\to c} f(x)=L$ then $\lim_{x\to c} 7f(x)=7L$

Prove that if $\displaystyle\lim_{x\to c} f(x)=L$ then $\displaystyle\lim_{x\to c} 7f(x)=7L$ I've never worked with limits, yet am trying to figure out how to prove this.
0
votes
1answer
71 views

Show that $4^\frac{1}{3}$ is an algebraic number?

How do you show that $4^\frac{1}{3}$ is an algebraic number? I don't understand the question nor how to begin on describing the proof to show what the question is asking.
3
votes
0answers
143 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...