# Tagged Questions

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### The limit of sequence of set.

When dealing with sequence of numbers, if $a_n\rightarrow a$, and for each $a_n$, we have $a_{n,j}\rightarrow a_n$, I think there exists a subsequence $a_{n,n_j}\rightarrow a$. What if we replace ...
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### Is there a straight-forward, “magic bullet” style way of showing $(\overline{X^\mathsf{c}})^\mathsf{c} = X^{\circ}$?

I would like to rigorously show that $(\overline{X^\mathsf{c}})^\mathsf{c} = X^{\circ}$, that is, the complement of the closure of the complement of X equals the interior of X. I am TAing a class ...
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### The product of two rational Dedekind cuts

If $a,b\in \mathbb{Q}$ and $C_a$ and $C_b$ are both positive rational Dedekind cuts then $C_a\cdot C_b=C_{a\cdot b}$. First of all this is my definition of product: Let $r,s$ Dedekind cuts such ...
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### How show $\mathbb N \cong \mathbb Q$ using Cantor pairing?

According to this: http://en.wikipedia.org/wiki/Cantor_pairing_function#Cantor_pairing_function, we can show that $\mathbb N\times\mathbb N\cong\mathbb N$. But as for $\mathbb Q$, this is not the ...
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### Countablity of the set of the points where the characteristic function of the Cantor set is not continous

We are creating the Cantor set typically starting from the interval $[0,1]$ and removing $\frac{1}{3}$ of it like it is described here or here. The problem is to resolve if the set of discontinuities ...
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### What is $\bigcup_{r\in(0,1)}[0,r]$?

Question: for any real number $r$, let $C_r$ be the closed interval $[0,r]$. Let $J$ be the open interval $(0,1)$. what is $\bigcup_{j\in J} C_j$? So far I have attempted a double inclusion proof to ...
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### Cardinality of a set of Continuous and Real Functions in the interval $[0,1]$

my question reads as: Let $\mathcal R[0,1]$ denote the set of all real-valued functions from $[0,1]$ to $\mathbb R$ and let $\mathcal C[0,1]$ denote the set of continuous functions on $[0,1]$. ...
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### My Proof for the Cardinality of a Particular Binary Distribution

my question reads as follows: I have constructed a proof and am concerned about 2 things: 1) The validity of my proof. 2) The construction of my proof. I am asking for someone to read through ...
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### Why there is not the next real number?

We can't say what is the just next real number (or rational or irrational number) of a given real number (or rational or irrational number respectively), what is the actual fundamental reason behind ...
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### Cantor Set Uncountability [duplicate]

The Cantor set is closed in $[0,1]$ and so its complement in $[0,1]$ should be a countable union of open intervals. Furthermore, every open set containing a point in the Cantor set contains a point ...
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### A Monkey Choosing Real Numbers for an Infinite Time

A common illustration of the nature of infinity is that, given an infinite amount of time, a monkey on a typewriter will, with probability $1$, produce the complete works of Shakespeare. Consider now ...
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### Show that the set is a singleton

Let $f$ be convex, differential function. Consider the set $$X=\left\{x\in \underset{x}{\text{argmin}} f(x):\; \|x\|\leq \|y\|,\;\forall y\in \underset{x}{\text{argmin}} f(x)\right\}$$ Prove that this ...
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### uncountably many numbers with lower bounded difference?

Does there exist an uncountable set U of reals s.t. $\forall a,b \in U\exists k>0$ giving $|a-b|>k$? This is impossible right? Because if not, then $\{(a,b)_{\lambda}\}$ will be uncountably ...
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### Real numbers in interval notation

If I want to denote the set $\mathbb{R}$ in $[a,b]$ interval notation, is it correct to say: $(-\infty, \infty)?$
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### What is the symbol $B_a$ mean in set notation

What is the symbol $B_a$ mean in set notation? $$\{Ba∣a∈A \text{and } B_a \text{ is the set of children of a}\}$$ I have seen the above used as an answer here. But I am not quite clear on what is ...
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### Check my proof of the “Boundedness theorem”

Theorem: Let $f$ be continuous on a closed interval $[a, b]$. Then f is bounded on $[a, b]$. Proof (sketch): Suppose $f$ is unbounded. Let's define the set $N$ containing those $x$ for which $f$ is ...
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### My proof of Bolzano's theorem

Before I read the proof of Bolzano's theorem from my Calculus book, I've tried to prove it myself. I will use the following lemma and the least upper bound axiom. [Lemma: Sign-preserving property of ...
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### Can I define sets as an infinite process?

Can I define sets this way in real analysis / set theory? I mean defining sets in kind of an infinite process and then taking their supremum or infinum. Let $S = \{1\}$. And for every element $x$ in ...
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### A basic property of the Lebesgue outer measure

If $G$ is a measurable set and satisfies $m^*(G)<\infty$, then for all $\varepsilon>0$ there exists a closed set $F\subset G$ such that $m^*(F)>m^*(G)-\varepsilon$ Edit: I know that: ...
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### How to prove that $\mathbb{Q}$ ( the rationals) is a countable set

I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it ...
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### Is it possible to form a bijection from N to Z if 0 is not an element of N?

I'm trying to show a bijection from $\mathbb{N}$ to $\mathbb{Z}$ for this assignment, which isn't terribly difficult however 0 is not included in $\mathbb{N}$ as defined by this book i'm working in... ...
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### Why does $\mathbb{R}$ have the same cardinality as $\mathcal{P}(\mathbb{N})$? [duplicate]

I recently read a fact that surprised me: The power set of the natural numbers is equinumerous with $\mathbb{R}$. In other words, $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}|$. I don't intuitively see ...
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### How do i do this process *precisely*?

Let $[a,b)\times [c,d)$ be a rectangle $R$ in $\mathbb{R}^2$. Let $\{[u_k,v_k)\times [p_k,q_k)\}_{1≦k≦n}$ be a mutually disjoint finite sequence whose union is $R$. Then we can decompose this into ...
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### Proving that a certain continuous function is surjective.

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $|f(x)-f(y)|≥|x-y| ,\forall x,y \in \mathbb R$ , then how do we prove that $f$ is surjective ?
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### Are there any known uncountable transfinite increasing sequences of real numbers?

The real numbers are uncountable, so assuming the axiom of choice there is at least transfinite sequence of real numbers $r_0, r_1, r_2, ..., r_\omega, r_{\omega + 1}, ...,$ up to (and possibly ...
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### Exercise Real Analysis

I'm having trouble to understand the following exercise I would appreciate any help? Let $A,B,C$ be sets such that $A\subseteq B\subseteq C$ and let $f:C\rightarrow A$ be an injective map. Define ...
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### Exponentiation as repeated Cartesian products or repeated multiplication?

In set theory, if $A$ and $B$ are sets, then their Cartesian product is defined to be $A\times B$ such that: $\forall x,y: [(x,y)\in A\times B \iff x\in A \land y\in B]$ Exponentiation (as repeated ...
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### Closed subset of R^2

Show that the set A = {(x,y): $x^3$ $>=$ $y^5$} is closed as a subset of $R^2$. I defined a closed set as a set whose complement is open. So the complement of the above set is {(x,y): $x^3$ ...
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### Bijectivity of set sequences

I've got this homework problem to prove in my introductory analysis course ... and right now, I really have no idea how to even go about that (and as such, don't really know the right questions to ...
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### The set of real numbers

I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these ...
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### $\mathbf{Q}$ is not a countable intersection of open sets $\rightarrow$?

I need help understanding a statement. I have been told that : $\mathbf{Q}\subset\mathbf{R}$ is not a countable intersection of open sets. In other words, ...
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### A finite set always has a maximum and a minimum.

I am pretty confident that this statement is true. However, I am not sure how to prove it. Any hints/ideas/answers would be appreciated.
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### Inequality with the supremum

I am trying to prove the following statement for $A\subset \mathbb{R},~\epsilon>0$ with $A$ bounded above: $\sup(A)-\epsilon<a\leq\sup(A)$, for some $a \in A$ I have tried dividing it into two ...
### Meaning of $\{ a,b \}$, and comparison with $(a,b)$
What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
I am reading a proof from a paper I found online, and it goes like this: We want to show that there exists a surjection $f$ from the cantor set $\mathfrak{C}$ to the interval $[0,1]$ then we can show ...