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

learn more… | top users | synonyms

1
vote
0answers
14 views

Negative of arbitrary lower bound is arbitrary upper bound?

Statement: Let $A \subset \mathbb{R}$ be bounded above. Prove that $\inf\{x\in\mathbb{R}: -x\in A\} = -\sup(A)$. Proof: Since $A$ is (implicitly nonempty and) bounded above, $\sup(A)$ exists. Denote ...
-2
votes
0answers
8 views

Questions involving work and time. [on hold]

I had this doubt since my 7 th grade that as to why is reciprocal taken in time and work questions but haven't got my answer.
0
votes
0answers
40 views

Rigorous Approach to Precalculus

I've made the mistake of looking at more advanced texts that deal with precalculus-level mathematics in a more formal, rigorous way than usual. Perhaps this isn't a mistake, but now that I've glimpsed ...
2
votes
2answers
31 views

What makes negative numbers different from positive numbers other than their being (almost) opposite?

To quote from Wikipedia's article on negative numbers Negative numbers represent opposites. If positive represents movement to the right, negative represents movement to the left. If positive ...
1
vote
1answer
28 views

Finding Limit Points?

I have two sets: $ A = ({{ (-1)^n + 2/n : n = 1, 2, 3, ...}}) $ and $ B = ( x \in \mathbb{Q} : 0 < x < 1 ) $ How does one go about finding the limit points for these sets? Would I just do $ ...
1
vote
2answers
25 views

Density of real numbers and density function

In Quantum mechanics, given a certain material, it is possible to write the density of energy states $\rho (E)$ as a function of $E$. That is: let's consider all the real values contained in the ...
0
votes
1answer
20 views

Proving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.

As the title says. I have to prove that the addition in the real numbers is well defined. Here are the definitions of both the real numbers and the addition of real numbers. These are translations. ...
0
votes
1answer
38 views

Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...
1
vote
0answers
43 views

The subset of $\mathbb R$ with $x\geq 0$ is closed and convex

Let $C=\{x\in \mathbb R|x\geq 0\}$. Prove that $C$ is a closed convex subset of $\mathbb R$ and show that for $x_0 \in \mathbb R$, the closest element in $C$ to $x_0$ is $max\{x_0,0\}$. I have looked ...
1
vote
3answers
48 views

Arithmetic Operations with Infinities in Real Analysis

Infinity is not a number , thus we cannot perform the usual arithmetic operations that we do with real numbers This is the usual reason given when asked why we can't perform the usual arithmetic ...
0
votes
0answers
21 views

Proof that the union of rational and irrational numbers sets is a set of real numbers [duplicate]

I see it all the time but is there a nice way to show that this is true? Or is this just a definition? I know that $\mathbb{Q} \subset \mathbb{R}$ and $\mathbb{I} \subset \mathbb{R}$, but how do we ...
3
votes
3answers
99 views

If $a$ and $b$ are roots of $x^4+x^3-1=0$, $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$.

I have to prove that: If $a$ and $b$ are two roots of $x^4+x^3-1=0$, then $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$. I tried this : $a$ and $b$ are root of $x^4+x^3-1=0$ means : $\begin{cases} ...
0
votes
0answers
20 views

Proof that there are no real zero divisors

I have went through a book that shows me the algebraic properties of $\Bbb R$ and I saw this valid logical equivalence: If $x,y\in\Bbb R$, then $x\bullet y =0$$\iff$ $(x=0)\lor(y=0)$ The way they ...
1
vote
0answers
24 views

Is there an application to NOT assuming that a square root is positive?

Further to the question here:Why is the even root of a number always positive? If it is mere "convention" (agreement) that we use positive real numbers as the even-powered-roots of positive real ...
2
votes
0answers
40 views

Does the exponent property $(a^{m})^{n} = a^{mn}$ break down in Real Analysis

Preface: I recently asked a question (The Definition of the Absolute Value), on the definition of the absolute value, in which I used $|x| = \sqrt{x^2}$, along with the exponent property $(a^{m})^{n} ...
0
votes
0answers
10 views

Least upper bound property of decimal representation of reals

This is my attempt at a proof that real numbers represented by infinite decimals satisfy the least upper bound property, i.e. every upper bounded set has a least upper bound. I am not sure it is ...
4
votes
1answer
66 views

Prove the existence of a real number satisfying a property

Let $x_1, x_2, . . , x_n$ real numbers from $[0, 1]$. Prove there is $x \in [0, 1]$ so that $|x - x_1| + |x - x_2| + . . . + |x - x_n| =\frac n 2$ My attempt Let $f:[0,1] \rightarrow R, ...
2
votes
1answer
60 views

Example of an uncountable subset of $[0,1]$ with no uncountable closed subsets

Question: Find an uncountable set $A \subseteq [0,1]$ s.t. for every uncountable subset $B$ of $A$, $B$ is not closed. Solution: My guess would be that $[0,1] \setminus \mathbb{Q}$ would be such a ...
2
votes
1answer
25 views

Limit points of $\lbrace f(n) \rbrace$

In this paper On the Limit Points of the Sequence $\sin n$ the author proves that the limit points of this sequence are the real numbers in $[0,1]$. My question is: Are there other functions $f$ such ...
0
votes
1answer
23 views

Construction a multiplicative inverse of a real number via Dedekind cuts

I'm reading Pugh's "Real Mathematical Analysis" where C.Pugh constructs real numbers system using Dedekind cuts. Unfortunately, he omits a construction of a multiplicative inverse of a real number $x ...
0
votes
0answers
37 views

Is this extension of the real numbers a field? It involves a unit of infinity.

Is this extension a field? Or perhaps some other structure? The extension depends on two basic ideas: A definition for a unit of infinity, the same as one given by Roger Penrose, and The infinity ...
2
votes
2answers
60 views

Random and non-computable numbers

Let $\alpha \in (0,1): \quad \alpha=0.a_1a_2\cdots a_n \cdots \quad$ where the $a_n$ are numbers generated by a physical generator of genuinely random numbers (if it exists). Than it seems that ...
1
vote
0answers
47 views

Can the sum of a finite series equal $\pi$?

Can the sum of a finite series equal $\pi$? I'm assuming of course that no element in the series is some fraction of $\pi$. I'm wondering since all methods I've seen of calculating $\pi$ involve ...
-1
votes
1answer
18 views

A question about the real line and the Dirichlet function.

Though the graph of the Dirichlet function is non-drawable, I think if we have to draw it in some informal way then it will be two complete lines (instead of isolated points). Here's my reasoning: ...
0
votes
2answers
42 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
vote
1answer
19 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?
2
votes
1answer
143 views

Which numbers are necessary?

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
votes
0answers
14 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
votes
1answer
53 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 ...
1
vote
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}, ...
0
votes
2answers
28 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
10 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 ...
0
votes
2answers
25 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
vote
2answers
71 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
240 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 ...
2
votes
2answers
45 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 ...
0
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 - ...
3
votes
1answer
34 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
112 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
votes
0answers
45 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
vote
3answers
30 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 ...
2
votes
5answers
80 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
70 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
29 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
vote
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
38 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
33 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
vote
2answers
32 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 ...