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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1answer
27 views

Is $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $f(\frac{a}{b}) = \frac{\max{(a,b)}}{\min{(a,b)}}$ a function?

Suppose that the relation $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $$ f \Bigl(\frac{a}{b} \Bigr) = \frac{\max{(a,b)}}{\min{(a,b)}} $$ is defined. Then is $f$ a function? If so, how would we prove ...
0
votes
1answer
21 views

Check my logical argument for this proof.

if x is a real number $x \not =\ 1 $, then there exists y which is also a real number $ ((y+1) \div ( y-2) ) = x .$ Prove it's converse also. Logical Argument: given: $x \not = 1$ Goal: $ ...
1
vote
0answers
20 views

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find $\overline A$, int$(A)$, and bdry$(A)$.

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find closure of $A$ $(\overline A)$, interior of $A$ (int$(A)$), and boundary of $A$ (bdry$(A)$). $A$ ...
2
votes
1answer
26 views

Set theory (containing Power Set) Need Help in a proof

I am confirming whether my proof is correct or not and need help. If $ A \subseteq 2^A , $ then $ 2^A \subseteq 2^{2^A} $ Proof: Given: $ \forall x ($ $ x\in A \rightarrow \exists S $ where $ ...
0
votes
2answers
33 views

How can I prove that $a_{n}$ and $a_{3n+2}$ converge to the same value?

I am given that $a_{n}\to L$, how can I prove that $a_{3n+2} \to L$ also? It makes sense since $3n + 2$ is still in $\mathbb{N}$, but I don't know how to say that in proof form.
3
votes
5answers
42 views

Sets $A,B,C$ with $B\subseteq C$, prove that $(A-B)-C=A-C$

Ran across this and couldn't figure out how you would give a formal proof. It seems intuitive, in that $(A-B)-C$ is the elements in $A$ but not in $B$, and then also remove the elements from $(A-B)$ ...
0
votes
2answers
46 views

Factorial Proof Problem

Suppose $m$ and $n$ are positive integers Prove $m!n! \lt (m+n)!$ I have something along the lines of: Since $1 \lt m+1$ and $2 \lt m+2$ etc.. then: $$n \lt m + n$$ So: $$n! \lt (m+n)!$$ I'm ...
2
votes
2answers
40 views

Show that $|e^z -1| \leq e^{|z|}-1$ for any z

Show that $|e^z -1| \leq e^{|z|}-1$ What i have tried is Let $z=x+iy$.Then, $$|e^z-1|=|e^x\cos y-1+ie^x\sin y|=\sqrt{(e^x\cos y-1)^2+(e^x \sin y)^2}=\sqrt{e^{2x}-2e^x\cos y+1}$$ I stuck here and ...
0
votes
1answer
20 views

Measure of open sets covering compact set

Prove that if $F$ is a finite collection of open intervals that covers a compact interval $[a, b]$, then the sum of the lengths of the intervals in the collection is strictly greater than $b − a$ ...
0
votes
1answer
40 views

Show $f(x)$ is bounded in a neighbourhood of its limit points

The attempt I made doesn't cover the case for $x=c$. How can I make it so it does? Prove that if a function $f : A \to \mathbb{R} $ has a limit $l \in \mathbb{R} $ at $c \in L(A)$, then it is bounded ...
0
votes
2answers
25 views

Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$

In a proof of Wilson's theorem, I read this identity and just wondered how to prove it: When $1\leq a\leq p-1$, we have $(a,p)=1$, so there exists a unique $\overline{a}$ with $a\overline{a}\equiv ...
1
vote
1answer
32 views

Showing $f$ is integrable on a plane, given a bound on its $L^{3/2}$ norm on certain regions

(old qual question in analysis) If $A_\lambda=\lbrace (x,y): \lambda \le x^4+y^2\le 2\lambda \rbrace$ and $f$ is locally in $L^{(3/2)}(\mathbb{R}^2)$ and there is an $a>3/8$, such that ...
2
votes
1answer
46 views

Induction Proof on String

Formally prove the correctness of the union construction as follows. Let $M_1$ and $M_2$ be the two $\lambda$-NFA's constructed for $R_1$ and $R_2$ and let $N$ be the $\lambda$-NFA constructed so ...
2
votes
2answers
42 views

Limit proof check, show $f$ is bounded in a neighborhood of its limit point

edit: as lem has pointed out, the case where x=c is not handled. Could someone suggest an idea? Prove that if a function $f : A \to \mathbb{R} $ has a limit $l \in \mathbb{R} $ at $c \in L(A)$, then ...
0
votes
0answers
33 views

Why does the Radius of Convergence prove the fundamental theorem of algebra?

The radius of convergence of $ \sum a_n (x-x_0)^n $ is given by $$ \frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| $$ if the limit exists in the extended reals. [Proof of ...
2
votes
4answers
62 views

Prove or disprove that the quotient ring is a field: $\displaystyle \frac{\mathbb Z_5[x]}{\langle 4x^3+ x^2+3\rangle}$

Prove or disprove that the quotient ring is a field: $$\frac{\mathbb Z_5[x]}{\langle 4x^3+ x^2+3\rangle}$$ Okay, so I need to either find an element that doesn't have an inverse or prove that they ...
2
votes
3answers
94 views

Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A = \sum_{i=1}^n f_0(x_1, \ldots, Ax_i,\ldots, x_m)$

Question: Consider $f: (-\epsilon, \epsilon) \to \mathbb R^{m^2}$ a differentiable path of matrices $m \times m$ such that $f(0) = I_m$ and the function $g: I \to \mathbb R$ is defined by $$g(t) = ...
2
votes
2answers
78 views

If $x$ is an integer then $x^2+ 5x - 1$ is odd.

What would be a proof strategy for this? I would like to show a proof of the contrapositive: if the expression is not odd, then $x$ is not an integer. If I go that route, how do I express the ...
1
vote
2answers
71 views

formula for the $n$th derivative of $e^{-1/x^2}$

$f(x) = \begin{cases} e^{-1/x^2} & \text{ if } x \ne 0 \\ 0 & \text{ if } x = 0 \end{cases}$ so $\displaystyle f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} ...
3
votes
0answers
24 views

Proof of equilateral triangle given angles

Let's say we start with a scalene triangle ABC, with no given angle measures or side lengths: Then, we add 3 Isosceles triangles adjacent to this one, given that they have angle measures ...
0
votes
2answers
33 views

Proof or a counterexample of a function

I have the following exercise, how can I proceed? Let $A$ and $B$ be sets, with $S \subset A$ and $f:A\to B$ a function, and $g:A\to B$ be an extension of $f\rvert_S$ to $A$. Does $g$ equal $f$? ...
1
vote
1answer
46 views

What is satisfiable by a reduct of a model is satisfiable by the original model (and vice versa)?

My professor told me that any formula that is satisfiable by a reduct of a model is satisfiable by the model it is a reduct of, and vice versa (as long as the formula is interpretable on the ...
4
votes
2answers
103 views

What is a good approach to demonstrate solvability of this type of puzzle without use of brute-force?

I chanced upon this puzzle in this question on the Anime & Manga site, and, like the OP, tried to solve it without any success. Here is a representation of the puzzle: the blocks may only be moved ...
0
votes
0answers
26 views

Help with solving ODE differently [duplicate]

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
1
vote
1answer
29 views

Contradiction in proof that in an integral domain, every prime is irreducible.

Let $\pi$ be a prime element in an integral domain. So, $\pi$ is a non-unit and if $\pi \mid ab \ $ then $\pi \mid a$ or $\pi \mid b$. An irreducible element $z$ is an element such that if $z=ab$, ...
0
votes
0answers
19 views

Locally injective function is globally injective [duplicate]

Let $f:\mathbb R\to \mathbb R$ be a continuous: Is the next statement true? If $f$ is locally injective for every real $x$ then $f$ is globally injective in $\mathbb R$ I think this theorem is true: ...
0
votes
2answers
23 views

Difference between open sets and open balls in metric space

Let $X$ be a separable metric space and let $\mathfrak{M}$ be the $\sigma$-algebra generated by open balls in $X$. Show that $\mathfrak{M}$ contains all the open sets in $X$ and all the closed ...
2
votes
3answers
38 views

Proving that $8^n-2^n$ is a multiple of $6$ for all $n\geq 0$ by induction

I have the following induction problem: $8^n-2^n$ is a multiple of $6$ for all integers $n\geq 0$. So far this is what I've done: Base case: $n = 0$ $8^0-2^0 = 6$ $1 - 1 = 6$ $0 = 6$ This ...
0
votes
2answers
64 views

Approaching this proof problem? If $0 \le x \le 3$ then $12 - 7x + x^2 \ge 0.$

Prove that if $x$ is a real number in the range $12 - 7x + x^2 \ge 0.$ Which type of proof should I use to solve this? At first I thought direct proof. Choosing a number between $0$ and $3$ and ...
3
votes
1answer
163 views
+100

Prove that there exists an $n\in\mathbb{Z}\cup\left\{-\infty,+\infty\right\}$ such that… (Dynamics)

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the following dynamics described by $T\colon X\to X$ as follows: A 1 becomes a 2, a 2 becomes a 0 and a 0 becomes a 1 if at least one of its two ...
0
votes
2answers
11 views

finding the derivative of w'Bw with respect to w

How should I approach to proving the following? $d(w^TBw)/dw = 2Bw$ where $B$ is a symmetrical matrix and $w$ is a vector.
2
votes
0answers
20 views

Prove that cube connot be tiled with $n>1$ cubes, such that all of them have different side length.

Prove that cube connot be tiled with $n>1$ cubes, such that all of them have different side length. I believe this is not hard problem, but I just do not have an idea how to start. I tried to ...
2
votes
4answers
112 views

Prove that $371\cdots 1$ is not prime.

Prove that $371\cdots 1$ is not prime. I tried mathematical induction in order to prove this, but I am stuck. My partial answer: To be proved is that $37\underbrace{111\cdots 1}_{n\text{ ...
1
vote
3answers
32 views

Help with discrete math proof

I'm having trouble with the following: $\ a_1=1$ and $a_n=1+\sum_{i=1}^{n-1} a_i$ for $n>1$ How should I go about proving the below? Any hints? $a_n = 2^{n-1}$
0
votes
1answer
12 views

Proving distance inequality between three elements in a normed linear space

For any two elements $x,y$ belonging to a normed linear space, distance between x and y is given by $\rho(x,y) = ||x-y||$ I am trying to prove the inequality $\rho(x,y) \leq \rho(x,z) + \rho(y,z)$ ...
0
votes
1answer
35 views

How to go about a “not divisible by..” proof

I need to show the following proof: For any integer x, x^2 + 4 is not divisible by 3. I was trying proof by contraposition, but I do not believe that is the most efficient way to go about this. ...
5
votes
2answers
59 views

How to decide which moduli to check when solving a “polynomial” congruence?

Consider the following problem: Find all integer solutions to $y^2 = x^5 - 4$. The solution goes something like – check modulo 11, where $x^5 \equiv 0, \pm 1$, and then check cases to arrive at ...
-3
votes
3answers
42 views

Stirling proof guidance [closed]

I would like some guidances: Prove that $$\lim_{n\to\infty}\frac{2^nn!}{n^n} = 0$$ Any help is greatly appreciate.
5
votes
4answers
75 views

Prove that if $A$ is both open and closed, $A=\mathbb R$. [duplicate]

Suppose $A$ is a non-empty subset of $\mathbb R$. Prove that if $A$ is both open and closed, $A=\mathbb R$. I think I'm supposed to assume that $A$ is not equal to $\mathbb R$ and derive a ...
1
vote
2answers
61 views

Prove by induction $n! > n^2$

I am trying to prove the inequality in the title for $n\geq 4$; however, I am stuck on the induction step! Any help would be appreciated. For $n\ge 4$, prove that $n! > n^2$. Base Case: $n=4$, ...
3
votes
1answer
42 views

If a sequence $f(x_n)$ goes to its minimum, will $x_n$ go to the point at which $f$ achieve the minimum?

I have a continuous function $f$ that is defined on a compact set. And $f(x_0)$ is its minimum. If I have a sequence $x_n$ such that $f(x_n)\to f(x_0)$, how can I show that $x_n\to x_0$? I tried ...
0
votes
1answer
21 views

Sum-of-divisors determinant

Let $\sigma_k(n)=\sum_{d|n}d^k$ be the generalized sum-of-divisors function. Let $S_n$ be the matrix defined by $[S_n]_{ij}=\sigma_i(j)$. I read a comment somewhere that $$\det(S_n)=1!\cdot 2!\cdots ...
2
votes
2answers
39 views

prove the nomalizer $N(H)$ of the subgroup $H$ in $G$ is a group

I need some help on the following question. For an arbitrary subgroup $H$ of the group $G$, the normalizer of $H$ in $G$ is the set $N(H) = \{x \in G \mid xHx^{-1} = H\}.$ Any help??
4
votes
4answers
85 views

Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically

I am working through the exercises and have spent half a day on one problem so I decided to get some help because I can't figure it out. Show that if $n$ is a positive integer at most equal to $m$, ...
0
votes
0answers
12 views

Prove that on N, the relation V where mV n

This is my question! Help me, please! Prove that on $\Bbb N$, the relation $\mathsf V$ is a linear order where $m\mathsf Vn$ if and only if $m$ is odd and $n$ is even, or $m$ and $n$ are even and ...
5
votes
3answers
122 views

Real Analysis book with pictures and ideas of proofs

I am taking real analysis course in my graduate class of Maths. My classes will start in 3 months. I have studied real analysis but not very rigorously. Whenever I see theorem I have no idea on how ...
2
votes
1answer
59 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 - ...
4
votes
3answers
79 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
22 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
vote
2answers
27 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 $$\sum_{1\leq n\leq x}\sum_{d\mid n}f(d)=\sum_{1\leq d\leq x}\sum_{1\leq m\leq x/d}f(d)$$ ...