For questions about the formulation of a proof, not about the mathematics behind it.

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3
votes
2answers
4k views

Prove that the min and max of 2 continuous function are continuous

Prove that if $f$ and $g$ are continuous functions the so are min⁡{f(x),g(x)} and max⁡{f(x),g(x)} I know this is true when $f$ and $g$ are not intersect each other, then I can compare them. However, ...
2
votes
3answers
44 views

Proof related to Harmonic Progression

The question is as follows: Let $m_1<m_2<m_3<\cdots<m_k$ be postive integers such that $\frac{1}{m_1}$, $\frac{1}{m_2}$, $\frac{1}{m_3}$, $\cdots$, $\frac{1}{m_k}$ are in arithmetic ...
0
votes
0answers
13 views

How to prove in $r_1p_1 +r_2p_2 =u\gcd(p_1,p_2)$, $u$ is a uniformly random polynomial.

Hypothesis: All polynomials are defined over a finite field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit prime number). Assume we have two fixed polynomials $p_1$ and $p_2$ of ...
3
votes
0answers
29 views

Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
0
votes
1answer
25 views

How to describe “the digits of n”

Prove that: 6 divides n if and only if 6 divides the sum of the digits of n Solution n mod 3 = 0 iff (sum of the digits of n) mod 6 = 0 How would one represent (the digits of n)?
3
votes
2answers
134 views

Prove that $(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$

For the non-negative real numbers $a, b, c$ prove that $$(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$$ What I did is applying Holder's inequality in LHS:$$(a^2+(\sqrt{2})^2)(b^2+(\sqrt{2})^2)(c^2+(\sqrt{2})...
0
votes
0answers
11 views

proof of preimage of union and intersection of sets

I was learning to proof the following proposition "the inverse image of an intersection or union equals the intersection or union of the inverse image" following these two really good youtube videos: ...
0
votes
0answers
54 views

Challenging Series example [duplicate]

Let $\{x_n\}$ be a decreasing sequence such that the series of $x_n$ converge. Show that the limit as $n$ approaches infinity of $\{nx_n\}$ equals zero.
1
vote
1answer
33 views

Box topology is finer than the uniform topology on $\mathbb{R}^\mathbb{N}$

This time, I wish to show that the box topology is finer than the uniform topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ However, the problem here is that ...
0
votes
1answer
37 views

How to construct a proof once we intuit a solution

For any integer N, there is an integer P such that one of the following is true: N = 10P N = 10P + 1 N = 10P + 2 N = 10P + 3 N = 10P + 4 N = 10P + 5 N = 10P + 6 N = 10P + 7 N = 10P + 8 N = ...
1
vote
1answer
26 views

Uniform topology is finer than the product topology on $\mathbb{R}^\mathbb{N}$

I wish to show that the uniform topology is finer than the product topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ We know both spaces are metrizable: The ...
0
votes
3answers
32 views

How to show that any separable space is CCC

I thought I had the proof of this in my head, but it doesn't sound right on paper. Can someone see if my argument could be improved. Let $(X,\tau)$ be a topological space that is separable, then it ...
1
vote
3answers
41 views

Show $d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$ is a metric on $C[0,1]$

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive ...
1
vote
1answer
35 views

Discrete Math Proof verification: products of floor

Determine if the following is true or false and provide a proof: $\forall x\in\mathbb{R},\exists y\in\mathbb{R}$ so that $\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor$ My attempt: -The ...
2
votes
1answer
35 views

Show that any metrizable space $X$ is Hausdorff

I wish to show that any metrizable space $(X,\mathcal{T})$ is Hausdorff Proof attempt: Let $d$ be the metric that generates the topology on $X$. Pick two points $x,y \in X$, we wish to produce two ...
1
vote
2answers
18 views

Show that any metrizable space $X$ is regular

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from ...
1
vote
2answers
80 views

Proving the Baire Category Theorem from scratch, Stuck!

Theorem: (Baire) $(X,d)$ is a complete metric space then the intersection of countably many dense, open subsets in the metric topology $\mathcal{T}$ generated by $d$ is dense In other ...
2
votes
2answers
81 views

Are theorems like subroutines for math? [on hold]

I've been developing more appetite for math just lately, as I study electromagnetics to deepen my understanding of electric circuits and devices. I'm finding that doing derivations as exercises helps ...
2
votes
0answers
34 views

Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
3
votes
1answer
65 views

proof of Triangle Removal Lemma

Where can I find a proof of the following version of Triangle Removal Lemma (or any version equal to it)? Let $G(V,E)$ be a graph on $n$ vertices such that it contains $\varepsilon n^3$ triangles, ...
2
votes
3answers
53 views

Proof by induction: inequality $n! > n^3$ for $n > 5$

I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
1
vote
2answers
41 views

Suppose X~Y, Prove that P(X) ~ P(Y)

My attempt: I imagined that if two sets are equivalent there would exist $ f:X→Y$ that is bijective. If I conceptually create P(X) and apply the function defined for the first equivalence relation to ...
2
votes
1answer
55 views

What does it mean in general to show something is well defined? [duplicate]

There is another post that addresses this but quickly fix the problem to be something in arthmetics, and in turn what it means for that arithematics problem to be well defined. I have never ...
0
votes
1answer
24 views

How to write this proof in a purely formal way?

Problem. Let $\Lambda$ be an index set. Then show that $$\displaystyle\bigcup_{\alpha\in\Lambda}\left(\displaystyle\bigcup_{B\in\gamma_{\alpha}}B\right)=\displaystyle\bigcup_{B\in\gamma}B$$where $\...
0
votes
1answer
57 views

Brute force way to show that $\rho(x,y) = \min\{1, d(x,y)\}$ is a metric

Following a hint in Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric I would like to use the brute force method to show that the standard bounded metric is a metric $$\rho(x,y) ...
0
votes
1answer
33 views

Proof idea: Let $(X,d)$ be a metric space, and $\rho$ be bounded metric, show that they will generate the same topology

Let $(X,d)$ be a metric space, $d$ generates the metric topology $\mathcal{T}$ via metric ball $B_\epsilon(x)$. Show that bounded metrics: $\rho_1(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ with ...
8
votes
2answers
5k views

Inductive Proof for Vandermonde's Identity?

I am reading up on Vandermonde's Identity, and so far I have found proofs for the identity using combinatorics, sets, and other methods. However, I am trying to find a proof that utilizes mathematical ...
50
votes
14answers
3k views

What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
1
vote
2answers
31 views

Trouble Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$

Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$ My problem with this question is that I do not know how one derives the theory in order to get the correct answer. I will ...
4
votes
1answer
588 views

extension of Cauchy's Integral formula

This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6. Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
1
vote
3answers
2k views

Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$

I have been hit with a homework problem that I just have no idea how to approach. Any help from you all is very much appreciated. Here is the problem Prove the equation: $a^{\log_b c} = c^{\log_b a}$ ...
1
vote
1answer
34 views

A simple proof in the form of an inequality

Proof that for all $a, b$ are elements of $\mathbb{R}$ : $(a+b)^2\geq 4ab$. Does it satisfies after doing some simple arithmetic to say that $(a-b)^2\geq 0$? Or do I need to go over all the cases ...
0
votes
2answers
56 views

Prove $\lim _{ n\rightarrow \infty }{ { x }_{ n }^{ k } } ={ \left( \lim _{ n\rightarrow \infty }{ { x }_{ n } } \right) }^{ k }$

I'm trying to prove that the limit of the sequence $x_n^k$ is the same as the limit of $x_n$ all raised to the $k$th power. Prove $$\lim _{ n\rightarrow \infty }{ { x }_{ n }^{ k } } ={ \left( \lim ...
1
vote
1answer
39 views

Fisher information for exponential family: Regularity conditions

for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus....
1
vote
0answers
29 views

How to calculate $\Delta$ in conic sections?

When learning conic section I learnt about $\Delta$. For any conic in general form : $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ Here $\Delta=abc +2fgh - af^2 - bg^2 -ch^2$ The conic is said to be ...
1
vote
2answers
43 views

If $\operatorname{Mod}(T_1 \cup T_2) = \emptyset$ then for some $\sigma$, $T_1 \vDash \sigma$ and $T_2 \vDash \neg \sigma$

Problem description: if $T_1$ and $T_2$ are theories such that $\operatorname{Mod}(T_1 \cup T_2) = \emptyset$, then there is a $\sigma$ such that $T_1 \vDash \sigma$ and $T_2 \vDash \neg \sigma$. I ...
0
votes
1answer
41 views

Proving the Mean Value Theorem

So I'm working through some questions in my book and I don't understand how to finish out the attached problem. For the first 5 blanks here is what I got: Since $f$ is continuous on $[16,25]$ and ...
3
votes
1answer
39 views

$f$ is injective if and only if for all $y \in B$ there exists at most one $x \in A$ such that $f(x)=y$

Let $A$ and $B$ be sets and let $f:A \to B$ be a function. Assume $f$ is injective. Let $y \in B$. There are two cases to consider. If there exists an $x \in A$ such that $f(x) = y$, then $x$ is ...
20
votes
14answers
2k views

How are proofs formatted when the answer is a counterexample?

Suppose it is asked: Prove or find a counterexample: the sum of two integers is odd The fact that 1 + 1 = 2 is a counterexample that disproves that statement. What is the proper format in which ...
0
votes
2answers
68 views

A **proof** for $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ [duplicate]

I need a proof for the inequality: $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ for all natural numbers $t \geq 2$. For $t=2$ both sides are equal. Can someone find a proof for all $t$? maybe ...
0
votes
2answers
76 views

Prove integral equality $ \int_{0}^{\pi} xf(\sin(x))dx = \pi \int_{0}^{\frac{\pi}{2}}f(\sin(x))dx $ [closed]

How can I prove the following claim for any given continues function: $$ \int_{0}^{\pi} xf(\sin(x))dx = \pi \int_{0}^{\frac{\pi}{2}}f(\sin(x))dx $$ Thanks!
0
votes
2answers
34 views

Hints on showing that a metric space is complete

Let $C[0,K]$ be the space of all continuous real valued functions on $[0,K]$ for $K>0$ and $L\geq0$, equipped with the metric $d$ defined by $$d(f,g)=\sup_{0\leq k\leq K}e^{-Lk}|f(k)-g(k)|.$$ I ...
11
votes
8answers
1k views

Book about technical and academic writing

I'm in the process of writing my Master's Thesis on automata theory. The writing must be in English which isn't my mother tongue. So the question is, given that this is my first time long (hundred ...
2
votes
8answers
229 views

Prove that $4$ divides $3^{2m+1} - 3$

Prove that $4$ divides $3^{2m+1} - 3$. By plugging in numbers I can see this is true, but I can't figure out a way to prove this, I was thinking maybe proving first that it is divisible by $2$, and ...
6
votes
1answer
565 views

Disproving existence statements

I am trying to get some practice on disproving existence statements and I was really stuck on this one: "There exists an example of three distinct positive integers different from $a,2a,3a$ for some ...
0
votes
3answers
44 views

Show that the closure of $A$ is the intersection of all closed sets containing $A$, topology proof needed

I want to show that given $(X, \mathcal{T})$, we define $\overline A = \{x \in X| \forall U \in \mathcal{T}, x \in U \implies U \cap A \neq \varnothing\}$ (definition of closure from Munkres), then ...
1
vote
1answer
47 views

Help in proof: a connected graph is $k$ edge connected iff all blocks are

Attempt: we know that the edge set of $G$ is the union of those of it's blocks (maximal connected subgraphs of $G$ not having a cut vertex), any two of them touching in at most one vertex. If all ...
6
votes
4answers
512 views

Proving a theorem, what is meant by sufficiency and necessity?

I am looking at the proof of a theorem and the proof begins by saying ...is the proof of the sufficiency part of this theorem so we just need to establish the necessity of the condition. What ...
0
votes
0answers
19 views

About continuity of scalar fields.

Using the usual definition of limits, with "epsilon and deltas", how can I show that if $x=(x_1,\dots,x_n)$ is a vector in $R^n$, and $f\colon J\to R$,where $R$ is the set of real numbers and $J$ is a ...
1
vote
1answer
34 views

Did I prove and disprove the following statements correctly?

Let $A = \left\{x \in \mathbb{Z} \mid \exists a\in\mathbb{Z}: x = 6a + 4\right\}$ and $B = \left\{y \in \mathbb{Z} \mid \exists b\in\mathbb{Z}: y = 18b - 2\right\}$ and $C = \left\{z \in\mathbb{Z} \...