# Tagged Questions

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

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### How to show that the set of all $(x,y)$ such that $3x^2 + 2y^2<6$ is an open set.

How can it be proved that the set of all $(x,y)$ such that $3x^2 + 2y^2<6$ is an open set? I tried to prove directly the aforementioned statement. Without success I tried to prove that the image ...
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### After the midnight when all trhee clock-hands are in same direction (superposed) again?

in order to solve this question: "After the midnight when all trhee clock-hands are in same direction (superposed) again?" that appears to be simple, probably is, but i could not give a answer ...
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### Prove that there exists n consecutive composite numbers

I'm asked to prove that there exists n consecutive composite numbers. This is what I've come up with. $$n! + 1 = (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot \dotsc \cdot n) + 1$$ is a prime number ...
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### Prove that $\lim \limits_{x \to 5}\left(4x^2-7\right)=93$

So I first need to determine the limit and then prove it: $\lim \limits_{x \to 5}\left(4x^2-7\right)$ So $L=93$ And thus $\left|f(x)-L\right|=\epsilon$ and $\left|x-c\right|=\delta$ Plugging ...
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### How to prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference?

how can i prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference? i mean supose that $A=(a_1,...,a_m)$ and $B=(b_1,...,b_m)$ both ...
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### Prove that if $f(x,t)$ is continuous in $D=\{(x,t):x\in[a,b]\land t\in[c,d]\}$ then $F(x)=\int_c^d f(x,t)\mathrm dt$ is continuous in $[a,b]$

Prove that if $f(x,t)$ is continuous in $D=\{(x,t):x\in[a,b]\land t\in[c,d]\}$ then $F(x)=\int_c^d f(x,t)\mathrm dt$ is continuous in $[a,b]$ This is about Riemann integration. I dont know how to ...
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### 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 ...
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### 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 ...
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### 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 $f(z)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $100$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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### 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?
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### 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 ...
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### 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 ...
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### 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,...
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### 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 ...
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: $$(1+q)(1+q^2)(1+q^4)...(1+q^{{2}^{n}}) = ... 1answer 59 views ### Proof that |S| \leq |T| if S \subseteq T. Let S and T be sets. I am having trouble proving that if S \subseteq T, then |S| \leq |T|, where |S| is the cardinality of S. 6answers 101 views ### Three positive numbers a, b, c satisfy a^2 + b^2 = c^2; is it necessarily true that there exists a right triangle with side lengths a,b and c? If so, how could you go about constructing it? If not, why not? I am new to proofs and I was reading a book where they posed this question. I understand that if we are given any right triangle, the ... 2answers 47 views ### (Real Analysis) Topology: Prove f(cl S)\subseteq clf(S) Let f:\mathbb{R}\rightarrow \mathbb{R} be continuous. Show: f(\overline{S})\subseteq \overline{f(S)} for S\subseteq \mathbb{R} (Note: \overline{S} denotes the closure of S; \partial S ... 2answers 54 views ### (Rigor/Validity of Proof) Every sequence of reals in a compact set has a convergent subsequence [ADDED/MODIFIED]: I began my proof with a compact set, but this was a wrong start. Although the comments are valid, I should've started with a bounded set. Because what I want to establish first is ... 2answers 20 views ### Strong Induction issue I am trying to prove a statement using strong induction but I seem to be getting stuck. I don't know if did something wrong or I am just not recognizing an opportunity for factoring/how to factor ... 2answers 65 views ### Proof that \overline{P(z)} = P(\overline{z}) for polynomial P with real coefficients Let$$ a_0, a_1, a_2, a_3, \ldots , a_n \quad (n \ge 1)$$denote real numbers, and let z be any complex number. With the aid of$$ \overline {z_1 +z_2+ \ldots +z_n} = \overline z_1 +\overline z_2+ \...
Suppose I know a result that the set of finite sets in $\mathbb{N}$ is countable. Is there a very quick way to show that the set of finite sets in any $X$ countable is countable? Idea...two sets ...