For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

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0
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2answers
37 views

Metric space on $\mathbb{R^n}$ where Heine-Borel criterion does not hold

Heine-Borel criterion of $\mathbb{R^n}$ : closed and bounded $\implies$ compactness Give an example of a metric space in $\mathbb{R^n}$ where this criterion does not characterize compactness ...
1
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1answer
18 views

Find the maximum value of the function

So I was just messing around with finding the maximum and minimum values of functions, and I came across this: $$ \text{Find the maximum value of} \,\, f(x)=\frac1{x^{2x^2}}.$$ Any ideas?
0
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0answers
55 views

Out of curiosity, which numbers are necessary? [on hold]

The Greeks were initially convinced that all numbers were rational until upon pain of contradiction were forced to accept that $\sqrt{2}$ was irrational and needed to be included in our number system ...
0
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0answers
13 views

Given 2 real numbers $a < b$ , let $d(x,[a,b]) = min\{|x-y| : a \leq y \leq b \}$ for $-\infty\leq x \leq \infty$

Then the function $f(x) = \frac{d(x,[0,1])}{d(x,[0,1])+d(x,[2,3])}$ satisfies (A) $0 \leq x < \frac{1}{2} $ for every $x$ (B) $0 < x < 1$ for every $x$ (C) $f(x) = 0$ if $2\leq x \leq 3$ ...
2
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1answer
41 views

A puzzle concerning the axiom of choice and the reals

Recently I was told the following riddle: Let $A=(a_1,...a_n,...a_{2n},a_{2n+1})$ a 2n+1-tuple of real numbers with the following property: Whatever number $a_i$ is removed from $A$ the remaining 2n ...
0
votes
0answers
32 views

Proving 0<1 without the use of contradiction

I am wondering if there exists a direct proof to the property of an ordered field that $$0<1$$ I've seen a proof by contradiction to this, but have yet to see a direct proof. Thanks!
1
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1answer
50 views

finding sup and inf of $\{\frac{n+1}{n}, n\in \mathbb{N}\}$

Please just don't present a proof, see my reasoning below I need to find the sup and inf of this set: $$A = \{\frac{n+1}{n}, n\in \mathbb{N}\} = \{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, ...
-9
votes
3answers
158 views

Please attempt to fault my proof of the continuum hypothesis [closed]

I have put this proof of the continuum hypotheses to both a Dr. of Maths and a Professor of Logic, and neither has demonstrated a flaw - although I doubt the professor (who shall remain nameless) ...
0
votes
2answers
26 views

$R/\Bbb Z$ isomorphic to $R/(2\pi \Bbb Z)$

I was told that $\mathbb{R}$$/$$\mathbb{Z}$ is isomorphic to $\mathbb{R}/2\pi \mathbb{Z}$ when these groups are taken under addition. Is this always true? I do not specifically see why this has to be ...
0
votes
1answer
9 views

Equivalence bounded limit superior

Suppose that $(x_n)_{n=1}^\infty$ is a real sequence such that $\limsup_n x_n$ exists. I wish to show that $\limsup_n x_n\le\beta \iff \forall\varepsilon>0\ \ \exists N\ \ \forall n\ge N, x_n ...
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votes
2answers
23 views

Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
0
votes
1answer
35 views

Does equality of the sum of two such series imply equality of each term of that series?

Let a(1)< a(2) < ..< a(m) and b(1)< b(2)<..< b(n) be real numbers such that $$\sum_{i=1}^m |a(i)-x| = \sum_{j=1}^n |b(j)-x|$$ for all x belonging to R. Show that m=n and ...
1
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2answers
69 views

How to define $[-\infty, \infty]$ or $[0, \infty]$?

I am familiar with basic undergraduate topology. For example, I know the process of one point compactification of a non-compact topological space, and how it applies to, say, $\mathbb R^2$. My ...
3
votes
2answers
235 views

prove a number is irrational [duplicate]

If $x$ and $y$ are irrational numbers then $x$ to the power of $y$ is irrational I am asked to prove or disprove this statement. To do so I got an idea to use the contra-positive, for that I need to ...
3
votes
2answers
41 views

Finding a term in a sequence

A strictly increasing sequence of positive integers $a_1, a_2, a_3,...$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the ...
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votes
0answers
24 views

How to discuss the continuity and differentiability of $f(x)$?

I have this problem on my Real Analysis problem set: Let $I_{A}(x)$ be the characteristic function of any set A. Consider $\begin{cases} f(x) = x^2 I_{\mathbb{Q}}(x)\\ g(x) = x^2 I_{\mathbb{R - ...
2
votes
1answer
32 views

Linear algebra with 2-dim. functions instead of matrices

I just thought about what would happen if we try to do matrix calculus with functions $\mathbb R^2 \to \mathbb R$ instead of matrices. The matrix multiplication would be something like $$ (f \times ...
6
votes
4answers
110 views

Prove this inequality: $\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3$

Let $a$,$b$,$c$ be positive real numbers such that $abc=1$. Prove that $$\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3$$ I tried various ...
0
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0answers
42 views

The set of perfect sets has cardinality $2^{\aleph_{0}}$

We say that $P \subseteq \mathbb{R}$ is perfect if it is closed and contains no isolated points. The claim is that $| \{ P \subseteq \mathbb{R} \mid P \text{ is perfect} \} | = 2^{\aleph_{0}}$. ...
1
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3answers
28 views

Does L'Hopitals Rule Work for Evaluating Difference of Limits?

Question : Given the function $f(x) = x^2 - e^x$, find the limit of $f$ as it approaches positive and negative infinity Finding the limit of $f$ as it approaches $-\infty$ is simple and the answer ...
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votes
0answers
24 views

What is probabilty of compund event from weights of basic events?

Suppose event occurs as combination of basic units . Now i assign weights to basic units such that they individual weights signify to what extent a basic unit will denote occurrence of event . Now I ...
2
votes
5answers
79 views

Cover $(0, +\infty )$ by open sets

Cover $(0, +\infty)$ by open sets $U_\alpha$ such that for any $\epsilon > 0$ there are points $x, y \in (0, +\infty)$ with $|x-y|<\epsilon$, not both belonging to the same $U_\alpha$ The ...
1
vote
2answers
68 views

Intuitionistic Real Analysis?

It seems like the following argument is in some sense the basis of real analysis: If $\forall\epsilon>0,\;\; d(x,y)<\epsilon$ then $x=y$. But in order to prove this statement, wouldn't ...
0
votes
1answer
28 views

Determine if $f=\{(x,y)\mid 2x+3y=7\}$ is invertible. From $\mathbb R \rightarrow \mathbb R$. If it is invert it.

I am thinking this is no, because I am not even sure if this counts as a function? I am unsure how this can be a function if there exist only a few $(x,y)$s that fulfill the equation. Or does the ...
1
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2answers
37 views

Expanding a Proof of Induction on $\Bbb N $ to $\Bbb Q $ (Linear Algebra)

My problem is the following: I have an $\Bbb R$ Vectorspace called $V$ and had to show via induction that $\langle nv, w \rangle=n \langle v, w \rangle$ for $v,w \in V$ and $ n\in \Bbb N$. (it's not ...
0
votes
1answer
37 views

Having trouble understanding how to disprove/prove if a formula is a function.

Is $\frac 1{x^2-2} $ a function from $\mathbb{R}\to \mathbb{R}$? Is it a function from $\mathbb{Z}\to \mathbb{R}$? I have been thinking about this but, I can't find any example for which you can have ...
2
votes
2answers
31 views

How can I check easily if all numbers in a number set equals to each other or not?

I want to know if all number pair equals each other that selected from a specified number set. For example: There is a set $A=\{5,3,6,2\}$ and to check the variable equalities requires to check the ...
1
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2answers
30 views

How to prove $(W_1\cap W_2)^{\perp}=W_1^{\perp}+W_2^{\perp}$ and $(W_1^{\perp})^{\perp}=W_1$ for the following condition?

Let $W_1$ and $W_2$ be subspaces of $R^n$, how to prove $(W_1\cap W_2)^{\perp}=W_1^{\perp}+W_2^{\perp}$ and $(W_1^{\perp})^{\perp}=W_1$? For the first one I have no idea how to get to the other side. ...
2
votes
1answer
61 views

Is there a “jagged” real-valued function that is “smooth” in cardinalities greater than the reals?

My background: I have a bachelor's CS degree and have never taken anything beyond part of a first course in abstract algebra - no real analysis or complex analysis. I learned about higher ...
1
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3answers
66 views

Is $\mathbb{R^2}$ Hausdorff? Give an example of a non-Hausdorff topology on $\mathbb{R}$

these are two questions on Hausdorff topological spaces. The bit I am having particular difficulty with is finding an 'example of a non-Hausdorff topology on $\mathbb{R}$' A Hausdorff topological ...
0
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1answer
64 views

Two problems with Exponents

How to solve following problems on exponents: $$\frac1{1+p^{a-b}+p^{a-c}}+\frac1{1+p^{b-c}+p^{b-a}}+\frac1{1+p^{c-a}+p^{c-b}}=?$$ and If $a^2bc^2=5^3$ and $ab^2=5^6$, what is $abc$? Please ...
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0answers
15 views

Modular arithmetic involving real numbers

For some $M \in \mathbb{N}$, let $a,b,r,s \in [0, M)$. Compute the following: $$ x = (a+r) \mod M$$ $$ y = (b+s) \mod M$$ $$ z = (x \cdot y) \mod M = (a \cdot b + a \cdot s + b \cdot r + r \cdot s) ...
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0answers
29 views

Why does a subfield of $\mathbb{R}$ not contain every real number?

Let $F$ be a subfield of $\mathbb{R}$. Since the completeness axiom holds in $\mathbb{R}$, it will also hold in $F$. But every real number can be represented as an infinite cauchy series of rational ...
1
vote
1answer
29 views

How to prove the following for an inner product space?

How to prove $\langle x,y\rangle=ax_1\bar{y}_1+bx_2\bar{y}_2$ defines an inner product of $\Bbb{C^2}\iff$$a,b\in\Bbb{R^+}$ $\Rightarrow$ if $\langle x,y\rangle=ax_1\bar{y}_1+bx_2\bar{y}_2$ defines an ...
0
votes
1answer
49 views

Is this a valid representation of real numbers?

I am trying to find the simplest representation of real numbers on the lambda calculus. I've thought about this one, and wonder if this is valid. First, we define a real number in the range ...
1
vote
1answer
55 views

Wouldn't real set be indeed smaller rather than bigger than the integer set? [closed]

This is linked to Natural and Real sets of numbers, which one is bigger than another? but clearly my question is of a different nature. Hello there, I would like to ask the community a question : ...
0
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0answers
35 views

Is this set of points dense in the closed interval?

Given a finite closed interval $[a, b]$ on the real line. If I define a set $S$ of real numbers such that $a$ and $b$ are in $S$ and for any two real numbers $x$ and $y$ in $S$, there is a real ...
58
votes
3answers
596 views

All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$

I would like some reference about this infinitely nested radical expansion for all real numbers between $0$ and $2$. I'll use a shorthand for this expansion, as a string of signs, $+$ or $-$, with ...
0
votes
1answer
45 views

Critique of proof for the square root of 2 being irrational

Is the following a valid proof that $\sqrt2$ is irrational? I've seen the proof to this in Baby Rudin, but I'm trying to figure out exactly how much "self expression" (I don't know what else to call ...
0
votes
1answer
33 views

Proof about infinite sum of fractions

STATEMENT: $t \in R$, $0 < t < 1$, $$\sum_{i=1}^\infty t^i = \frac{t}{1-t}$$ EXAMPLE: let t = $\frac{1}{2}$ ...
0
votes
1answer
41 views

Proof for divisibilty tests for 13, 16, 17,19

I would like to know the divisibility tests for 13, 16, 17, 19. I also would appreciate the proof for the divisibility test done. Please oblige! Rgds Jayanth
2
votes
1answer
34 views

Decimal expansions and topological connectedness

I'm a bit confused by the following footnote from Moschovakis's Notes on Set Theory, p. 135fn24 (in the note, $\mathcal{N}$ denotes the Baire space). The puzzling part is in bold: One may think of ...
1
vote
1answer
50 views

Why there are real numbers with infinite digits, but no such natural numbers (or another reason why real numbers are uncountable)

This question is me trying to understand (again) why there can be no one-to-one correspondence between the sets of natural and real numbers. The source of confusion is this: if we abstract completely ...
0
votes
1answer
36 views

Proving the arithmetic-geometric mean inequality using just only the field and order axioms

Using just the axioms, prove the arithmetic-geometric mean inequality: $$\sqrt{ab}\leq\frac{a+b}{2}$$ for any $a, b \in\mathbb R$ with $a > 0$ and $b > 0$. (Assume, for the moment, the ...
0
votes
1answer
47 views

Proof help number theory

Let $A\subseteq\Bbb R$ be nonempty. If $\sup(A) \in A$ then $\sup(A)$ is the largest element of $A$, i.e., $\sup(A) = \max(A)$. Conversely, if $A$ has a largest element then $$\max(A) = \sup(A)$$ ...
1
vote
2answers
57 views

show that this statement is false (counterexample) if $a,b \in \mathbb R \backslash \mathbb Q $ then $a \cdot b \in \mathbb R \backslash \mathbb Q $

if $a,b \in \mathbb R \backslash \mathbb Q $ then $a \cdot b \in \mathbb R \backslash \mathbb Q $ Okay so the question asks to show, with a counter example, that the above statement is false. Here ...
0
votes
0answers
15 views

Help proving $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $

I am trying to formally prove: $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $ where n is an integer, and x and y are natural numbers. It is obvious that, when $\frac xy$ is ...
7
votes
5answers
656 views

Is this direct proof of an inequality wrong?

My professor graded my proof as a zero, and I'm having a hard time seeing why it would be graded as such. Either he made a mistake while grading or I'm lacking in my understanding. Hopefully someone ...
0
votes
0answers
37 views

Elegant proof, that the function is not nonnegative defined

Suppose that $\varphi:\mathbb{R}\to\mathbb{C}$ is an even function, i.e. for all $\lambda \in \mathbb{R}\,\,\,$$\phi(\lambda)=\phi(-\lambda)$. We say that $\varphi$ is non-negative (positive) ...
3
votes
0answers
37 views

Rational numbers as angles - where do irrationals fit in?

If we make a rectangular grid with integer coordinates, it's possible to assign a unique angle to any rational number, using the definition $\tan \phi=y/x$ for $\phi \in (-\pi/2, \pi/2)$. For ...