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

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2
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1answer
18 views

Proof verification about inverses of linear mappings.

In order to prove the following statement: "Let $F: U\to V$ be a linear map, and assume that this map has an inverse mapping $G\colon V \to U$. Then $G$ is a linear map." In Serge Lang book he ...
2
votes
2answers
4k views

Proof of Bezout's Lemma using Euclid's Algorithm backwards

I've seen it said that you can prove Bezout's Identity using Euclid's algorithm backwards, but I've searched google and cannot find such a proof anywhere. I found another proof which looks simple but ...
0
votes
1answer
403 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
0
votes
1answer
32 views

Proving that the following is a convex subset

It is given that $I(S)=\{z|\exists x,y \in \mathbb{R}$ such that $(x,y,z)^t \in S$}$\subseteq \mathbb{R}$. How do I show that I(S) is a convex subset? We know that $S \subseteq \mathbb{R}^3$ is ...
0
votes
3answers
720 views

Proving the Division Algorithm using induction

Let $n \in \mathbb{N}$. For every $m \in \mathbb{Z}$, there exist unique $q, r \in \mathbb{Z}$ such that $ m = qn+r$ and $0 \le r \le n-1$. We call $q$ the quotient and $r$ the remainder when dividing ...
2
votes
1answer
117 views

proof of$\frac{\partial^2 f(x,y)}{\partial x\partial y}$=$\frac{\partial^2 f(x,y)}{\partial y\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial^2 f(x,y)}{\partial x\partial y}=\frac{\partial^2 f(x,y)}{\partial y\...
1
vote
1answer
24 views

Primitive = Non-negative + Irreducible + 1 Positive element on main diagonal

Can anyone provide me with the proof for the sufficient condition for a matrix to be primitive as described by the definition from planetmath.org? (http://planetmath.org/primitivematrix)
0
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1answer
35 views

Proof of non-constant analytic functions

Use the following theorem: "A function that is analytic in a domain $D$ is uniquely determined over $D$ by its values in a domain, or along a line segment, contained in D" to show that if $f(z)$ ...
1
vote
1answer
28 views

Proof of Reflection Principle when f(x) is imaginary

Suppose that a function f is analytic in some domain $D$ which contains a segment of the x-axis and whose lower half is the reflection of the upper half with respect to that axis then $$\overline{f(z)...
1
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4answers
37 views

Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric.

How can i proof the following statement: "Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric." i tried to work out the properties of a matrix to be ...
0
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1answer
34 views

How can i proof that every high order derivative of $\frac{1}{1+x}$ is equal to$ (-1)^kk!$ at point $0$.

In order to calculate the Taylor-Maclaurin polynomial $\frac{1}{1+x}$ of order $n$ at point $0$, i used the identity: $$\sum_{i=0}^n x^i + \frac{x^{n+1}}{1+x} = \frac{1}{1+x}\qquad (I)$$ and i ...
-2
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0answers
52 views

Can we derive $ax^3+bx+c$ in a similar fashion? [on hold]

Derive the formula for $\color{orange}{ax^2+bx+c}=0$ Divide by a: $x^2+{b\over a}x+{c\over a}=0$ Note that: $\color{red}{(x+{b\over 2a})^2}=x^2+{b\over a}x+({b\over 2a})^2$ Add $({b\over 2a})^2$: $\...
0
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0answers
28 views

Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
0
votes
1answer
21 views

Prove if $x_1,…,x_n$ are natural numbers with $n\geq2$ then $x_1x_2…x_n$ is odd iff $x_i$ is odd for all $i$, $1\leq i\leq n$

I am not sure if Im on the right track here but if any one could help out I would greatly appreciate it. Prove if $x_1,...,x_n$ are natural numbers with $n\geq2$ then $x_1x_2...x_n$ is odd iff $...
0
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0answers
28 views

How to do proofs involving sets

I have just recently started preparing for a course I will be taking next year, but I have very limited knowledge as it relates to proofs. It seems as though the only proofs I am slightly familiar ...
1
vote
1answer
28 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
0
votes
2answers
174 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
1
vote
3answers
98 views

Book Recommendations for Writing Proofs

As an applied mathematics student coming from a small university, I have not had an adequate course in writing/formulating proofs for problems in advanced calculus/real analysis (my university has an ...
1
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1answer
53 views

Formulating a problem in terms of set theory

Here is one problem I was trying to solve just by trial-and-error method. However, I was thinking about how to write the clear solution using set theory. Problem: A notebook contains exactly $100$...
0
votes
2answers
37 views

binomial inequality with sums

Assume I have a series of numbers $a_1 \dots a_n$ where $0 \leq a_i \leq n-1$ and a positive integer $r$. how to show that the sum of number of ways to choose $r$ from $a_i$ is at least as $n$ times ...
2
votes
4answers
6k views

Proof by induction that $ \sum_{i=1}^n 3i-2 = \frac{n(3n-1)}{2} $

I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically: $$ \sum_{i=1}^...
1
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2answers
47 views

Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
0
votes
0answers
31 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
1
vote
1answer
35 views

Is it sufficient to prove that a function is an open map by looking at the basis element?

I am trying to prove that the projection map $\pi_X:(X, T)\times (Y,J) \to X$ is an open map But I don't know if I can use the basis element directly, so my proof is quite round about and lengthy ...
0
votes
1answer
855 views

Disjunctive normal form (BOTH dnf and cnf) example help

T or ( not x and y ) or ( x and y ) In class we went over examples of how many things can be both DNF and CNF... eg. (not z) or y can be thought of as both (not z) or y .... dnf ((not z) or y)...
1
vote
1answer
15 views

Show that there exists at most one extension of $f$ whose co-domain is a Hausdorff space [duplicate]

I want to show the following Suppose $A \subset X, f: A \to Y$ is continuous, $Y$ is Hausdorff. Show that there is at most one continuous extension $g: \overline A \to Y$ I feel like I am ...
1
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2answers
44 views

How to see that $f(t) = (t, 2t, 3t, \ldots)$ continuous in the product topology

I am trying to check whether $f: \mathbb{R} \to \mathbb{R}^\omega$ $f(t) = (t, 2t, 3t, \ldots)$ is continuous or not in the product and box topology. But I have a feeling I don't have the ...
3
votes
3answers
68 views

Topology: is it ever good to write $x \in U \in \mathfrak{T}$

Sometimes I come across a sentence in my topology book that says, let $U$ be an open set that contains $x$ I can't help but write it down as: Let $$x \in U \in \mathfrak{T}$$ Is it good ...
0
votes
0answers
25 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....
3
votes
4answers
93 views

$a^n$ even implies $a$ even

I've tried to prove that $(\forall a,n>0 \in \mathbb{N}),(a^n \text{ even} \implies a \text{ even})$, can someone tell me whether my proof is sound? Lemma 1: $a \text{ even} \implies a^2 \text{ ...
2
votes
0answers
51 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, ...
1
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2answers
54 views

Prove that the directrix-focus and focus-focus definitions are equivalent

(NOTE: This is my attempt at answering this question and this question, but I rewrote it in order to make it easier for me to solve. Also, I've made a YouTube video explaining this whole proof. Note ...
0
votes
1answer
13 views

If unions of two families sets are disjoint then families of sets are disjoint too.

I have read that theorem "Suppose $\mathcal{F}$ and $\mathcal{G}$ are families of sets. If $\cup\mathcal{F}$ and $\cup\mathcal{G}$ are disjoint, the so are $\mathcal{F}$ and $\mathcal{G}$" is ...
3
votes
1answer
120 views

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable?

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable? To clarify: the problem stated that the composition is well ...
2
votes
2answers
43 views

Proof verificication and question of rigour: $A$, $B$, connected implies union is connected

Don't mark this as duplicate. The other question does not help me figure out how rigorous MY proof is. Problem: Let $A$ and $B$ be connected subsets of a metric space and let $A\cap\overline{B}\neq\...
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votes
3answers
73 views

prove that the $5$th root of $r$ is irrational if $r$ is irrational [closed]

I am trying to learn mathematics for computer science in own efforts. I got this task to prove that $\sqrt[5]{r}$ is irrational, given that $r$ is irrational. Normally if I had to prove that $\sqrt{2}...
0
votes
1answer
43 views

Nice way to prove a limit.

I know how to prove the following limit $$\lim _{\epsilon \rightarrow 0} \frac{a^{\epsilon}-1}{\epsilon}=\ln(a)$$ But I am looking for a nice way to do it, a little elegant. Would you have one?
1
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3answers
60 views

Prove $\forall n \in \mathbb{N}$ where $ n \neq 1 , n + \frac{1}{n} > 2$ using completing the square.

I have got this far; I am only unable to understand how to finish the proof. $n>0 \implies n + 1/n > 0 \implies n + 1/n + 2 - 2 > 0 \implies {\big(\sqrt{n}+\frac{1}{\sqrt{n}}\big)}^2 - 2 >...
4
votes
1answer
1k views

If $\lim f(x) = 0,$ then $\lim 1/|f(x)| = \infty.$

The Problem: Suppose that $f:D\to\Bbb R$, where $D$ is a subset of $\Bbb R$ and $a$ is an accumulation point of $D$, $\lim_{x\to a}f(x)=0$, and $f(x)\ne0$ for any $x$ in $D$ in some neighborhood ...
30
votes
9answers
3k views

Why don't Venn diagrams count as formal proofs?

Just curious. If the purpose of a proof is to inform and persuade, why don't Venn diagrams count? Is it just convention or is there a more, umm, formal reason haha. Thanks!
2
votes
8answers
200 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 ...
1
vote
2answers
32 views

Show that a set is open if and only if each point in S is an interior point.

I am in a complex analysis class and have been asked to prove this. I know I have to prove both ways so. If a set is open then each point in $S$ is an interior point. Proof: Let $S$ be an open set,...
2
votes
6answers
100 views

Three positive numbers a, b, c satisfy $a^2 + b^2 = c^2$; is it necessarily true that there exists a right triangle with side lengths a,b and c?

If so, how could you go about constructing it? If not, why not? I am new to proofs and I was reading a book where they posed this question. I understand that if we are given any right triangle, the ...
2
votes
1answer
97 views

How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? [duplicate]

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function... I tried my hand at the ...
4
votes
2answers
88 views

problem proving: $(1+q)(1+q^2)(1+q^4)…(1+q^{{2}^{n}}) = \frac{1-q^{{2}^{n+1}}}{1-q}$

I'm trying to prove this, and it is really frustrating, because it seems a really easy problem to prove, however, I'm having a little problem with exponents: $$(1+q)(1+q^2)(1+q^4)...(1+q^{{2}^{n}}) = ...
0
votes
1answer
58 views

Proof that $|S| \leq |T|$ if $S \subseteq T$.

Let $S$ and $T$ be sets. I am having trouble proving that if $S \subseteq T$, then $|S| \leq |T|$, where $|S|$ is the cardinality of $S$.
2
votes
3answers
38 views

Principle of Mathematical induction proof

Prove that $2^n >n$ for all positive integer $n.$ I know this can be easily proved by using PMI Let $P(n): 2^n > n$ For $n = 1$ $$2^1 > 1.$$ Hence $P(1)$ is true. Assuming that $P(k)$ is ...
1
vote
6answers
163 views

Proving that $64$ divides $3^{2n+2}+56n+55$ by induction

Let $n ≥ 0$ be an integer. Prove by induction: 64 divides $3^{2n+2} + 56n + 55$ general expression: $3^{2n+2} + 56n + 55 = 64m$ 1st I substitute $P(0)$ and it gives me true: $9+55 = 64$ (if m = 1 ...
0
votes
2answers
46 views

(Real Analysis) Topology: Prove $f(cl S)\subseteq clf(S)$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Show: $f(\overline{S})\subseteq \overline{f(S)}$ for $S\subseteq \mathbb{R}$ (Note: $\overline{S}$ denotes the closure of S; $\partial S$ ...
3
votes
1answer
781 views

Counting Elements and Their Inverses

The problem I am attempting to prove is the following: In any finite group $G$, the number of elements not equal to their own inverse is an even number. Caveat: I have had very limited experience ...