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

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-4
votes
1answer
39 views

Add or subtracts 1 to 9 numbers and get the answer 100 [on hold]

so the question is that we have add or subtract numbers from 1 to 9 and the answer should be 100. (Note: The numbers shouldn't repeat). so what is the solution to this problem? Please answer this ...
2
votes
5answers
75 views

Question about using arbitrary $\epsilon$ in real analysis proofs

I've notice that in a lot of the proofs that are assigned in an undergraduate analysis course, we are often trying to show that some quantity is bounded by an arbitrary epsilon. For example, if I ...
4
votes
2answers
68 views

$\sum_{n=1}^{\infty} \frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)$ diverges [on hold]

How can I prove the divergence of the series $$\sum_{n=1}^{\infty} \left(\frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)\right) $$ if $f:\mathbb{N} \rightarrow \mathbb{N}$ is injective? $ $
1
vote
2answers
103 views

What is the sum of all real numbers from $0$ to $1$? [on hold]

I wanted to know the approximate sum of real numbers from 0 to 1. Please tell me how we can find it.
-4
votes
1answer
41 views

How to find the L.C.M. of $\frac{15}{2}$, 6 and 5? [on hold]

I am unable to find the L.C.M. of $\frac{15}{2}$, 6 and 5? I have never dealt with L.C.M. of fractions. I tried to find the L.C.M. by making $\frac{15}{2}$ decimal, but in vain. Any hint would be ...
1
vote
0answers
39 views

What logical resources does one need to distinguish $\mathbb{R}\hspace{-0.05 in}-\hspace{-0.04 in}\mathbb{Q}$ from $\mathbb{Q}$?

(I'm inspired by this question.) As a strict lower bound, having just the order relation is not enough, since $\mathbb{R}\hspace{-0.05 in}-\hspace{-0.04 in}\mathbb{Q}$ and $\mathbb{Q}$ are both ...
3
votes
1answer
64 views

Showing $\sqrt{2}, e, \pi$ are real numbers in the axiomatic approach to defining $\mathbb{R}$

I would appreciate if someone could demonstrate how to show $\sqrt{2}, e, \pi$ are real numbers in the axiomatic approach to defining $\mathbb{R}$ (without reference to a model). The Wikipedia page ...
5
votes
1answer
71 views

Inequality with reciprocals of $n$-variable sums

Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Is it always true that $$\sum_{i=1}^n\frac{1}{a_i}-\sum_{1\leq i<j\leq n}\frac{1}{a_i+a_j}+\sum_{1\leq i<j<k\leq ...
12
votes
8answers
758 views
+50

When trying to learn analysis from bottom up, what numbers should I first construct?

I am interested in studying more analysis and related topics. However, I want to make sure I do so well and without making too many broad jumps in my learning. In some books I have seen, the author ...
0
votes
1answer
34 views

Looking for Clarification on a proof of Density of Q in R

I am looking for some advice/help in regard to the proof that Q is dense in R, given in Walter Rudin's book "Principles of Mathematical Analysis". Mostly, I want to see if my reasoning is correct for ...
1
vote
1answer
80 views

Question regarding complex numbers and real numbers?

I have two questions... If we take $(-1/3)^{(-1/3)}$ it would equal $-1.44224957$ since... $$(-1/3)^{-1/3}$$ $$\frac{1}{(-1/3)^{(1/3)}}$$ $$\frac{1}{-0.6933612744}$$ $$-1.44224957\ldots$$ Yet when I ...
2
votes
2answers
360 views

Does the following region have positive area?

Does the following region have positive area? $\{(x,y); x>0\}$. Now we know the area is infinite. But is it wrong if I were to say that it does have positive area?
1
vote
2answers
42 views

Union of some open balls in $\mathbb{R}^2$

Let $B_n$ denote the open ball in $\mathbb{R}^2$ of radius $n$ centered at $(n,0)$. $$A=\bigcup_{n\in \mathbb{N}} B_n$$ Show that $A$ is the open right half plane. i.e. $A=\{(x,y) \in \mathbb{R}^2 ...
0
votes
1answer
17 views

About the solvability of certain equations

How one can see if this equation has real solutions: $$x^{2^{k}}-x-a=0$$ where $x$ is the unknown, $k$ is a positive integer and $a$ is a real constant.
1
vote
1answer
46 views

What is the Lebesgue measure of a circle in $\mathbb{R}^2$

What is the Lebesgue measure of a circle in $\mathbb{R}^2$ So my answer of zero. The way I do it is by putting the circle inside an annular ring and then shrinking the ring so that its measure (which ...
1
vote
0answers
173 views

Very simple proof that $\sqrt{2}$ is irrational.

I came across a nice-looking proof that $\sqrt{2}$ is irrational here. It somehow seems to good to be true. What are the assumptions being made in the proof and if this proof is indeed correct, why is ...
3
votes
2answers
151 views

Why is the observation that proof of the Fundamental Theorem of Algebra requires some topology not tautological?

I have heard it mentioned as an interesting fact that the Fundamental Theorem of Algebra cannot be proven without some results from topology. I think I first heard this in my middle school math ...
2
votes
1answer
63 views

Rudin's definition of derivative

Walter Rudin's Principle of Mathematical Analysis defines the derivative as follows in Definition 5.1: Let $f$ be defined (and real-valued) on $[a,b]$. For any $x \in [a,b]$ form the quotient ...
1
vote
2answers
50 views

Prove that if $0\leq a < b$ and $0 \leq c <d$, then $ac<bd$

Proof: Suppose $0\leq a<b$ and $0\leq c < d$. Let $P$ denote the set of positive real numbers, we see that $$a, b-a, c, d-c \in P.$$ By closure under multiplication, $$(b-a)(d-c) = ...
2
votes
3answers
97 views

Show that $\mathbb{Q} \subset \mathbb{R}$, given the axiomatic definition of $\mathbb{R}$

If instead of constructing $\mathbb{R}$ from $\mathbb{Q}$, we define it axiomatically, as a complete ordered field, how would one show that $\mathbb{Q} \subset \mathbb{R}$?
0
votes
1answer
41 views

What is the limit of the ratio of the sum of all real numbers from 0 to 2 over the sum of all real numbers from 0 to 1.

I approached this question I made, by saying that the sum of a finite amount of numbers from a to b separated by a common change is average•(# of numbers we have). I found the formula the # of the ...
3
votes
2answers
95 views

Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$

Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$ Values $a,b,c$ are all positive reals. I tried Muirhead and a few AM $\geq$ GM. This problem is equivalent to proving $a^4b^3 + b^4c^3 ...
0
votes
1answer
15 views

Are all piecewise functions formed of finite pieces when domain restricted to an interval?

I'm working with infinite pieces piecewise functions with real domain and co domain. I'm wondering if there is any function that might be definable that will not have finite pieces when domain is ...
3
votes
2answers
48 views

Math Homework Question

I think I might be overthinking this question and I need some help Assume only the knowledge of addition and multiplication of real numbers. What do we mean by $u^{-1}=v$ ? Can this equality be ...
1
vote
2answers
39 views

What is the portion of an integral solution after the comma?

I'm trying to brush up on my Calculus (took it 15 years ago) just for a little fun. I solved the following integral, given as a sample "easy" integral by a buddy who is still up to snuff on Calc: ...
0
votes
4answers
14 views

An inequality, comprising of liminf and Sup of a set

The following was an example in a book called Applied analysis by Hunter. I am not exactly sure of what it really means or to how to approach it { $ x_{n,\alpha}\in \mathbb R $ |$ n \in \mathbb N $ ...
1
vote
2answers
70 views

Version of the Axiom of Induction for Real Induction?

Mathematical induction can be done using the axiom of induction, which is given as a formula written in the language of mathematical logic. Is there a way to express the ideas behind 'real induction' ...
0
votes
0answers
16 views

How to compute the logarithm of a computable number

Let's say you have a computable number $x>0$. $\ln(x)$ is computable as well. Given the computability of $x$, what is a computation for $\ln(x)$. I am using the definition where $a$ is computable ...
0
votes
1answer
26 views

Axiomatic proof help?

Can anyone help me to prove this using only the axioms? " If a is a positive real number which is less than the real number b, then the negative of the reciprocal of a is less than the negative of ...
-1
votes
2answers
33 views

Exponentiation of real numbers [closed]

If $x>0$ is a real number such that $x^\alpha<1$ for some real number $\alpha$. Then $x<1$? Thank you.
1
vote
2answers
52 views

Find all values of $n$ for which $n^2+96$ is a perfect square

There can be infinite values of $n$. The statement is true for $n=2,5$. How to find out others? Please tell if there is any formula.
1
vote
2answers
62 views

find the last digit of the sum $1!+2!+3!+…+49!$ [closed]

Is there any formula for finding the last digit of the factorials? How to approach these type of questions? Thanks in advance.
6
votes
1answer
73 views

About $\mathbb{R}$ as a vector space over $\mathbb{Q}$

I want to understand better the structure of the vector space $\mathbb{R}$ over $\mathbb{Q}$. I know that it is an infinite dimensional vector space with a non countable Hamel basis, and it is cited ...
12
votes
1answer
110 views

What do you get with this equivalence relationship for all $\mathbb{Q}$ sequences

Consider all $\mathbb{Q}$ Cauchy sequences with this equivalence relationship $\{x_n\} \sim \{y_n\} \iff \{x_n-y_n\} \rightarrow 0$ Then you get all real numbers as an equivalence class with this ...
0
votes
0answers
29 views

Finite Interval under Nonnegative Real Numbers

I want to construct a generic finite interval under $\mathbb{R_+}$ - it should be bounded, closed and it should include $0$ as the lower bound. This would allow me to choose an element from this ...
7
votes
2answers
83 views

Clarify: “$S^0$, $S^1$ and $S^3$ are the only spheres which are also groups”

The zero, one, and three dimensional spheres $S^0$, $S^1$ and $S^3$ are in bijection with the sets $\{a\in \mathbb{K}:|a|=1\}$ for $\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}$ respectively. The ...
0
votes
0answers
40 views

Linear independence of real powers of x

I know that integer powers of x are linearly independent. I would expect that fractional powers of x (eg. $x^{n/2}$) are also linearly independent. But what about real powers of x? If I could use any ...
2
votes
3answers
35 views

Intuition for interior points and non-open sets

I'm currently reading a book on real analysis. There is a statement right in the introductory section which I'm struggling to intuitively understand. It goes as follows: Assume a sequence $(O_n)$ ...
0
votes
1answer
47 views

A question on Dedekind's cuts

Rudin write that the members of $\mathbb R$ will be certain subsets of $\mathbb Q$, called cuts. A cut is, by definition, any set $\alpha\subset \mathbb Q$ with the following three properties. (I) ...
11
votes
6answers
2k views

Is there a real number which comes just after a particular real number?

This is like when we say that the integer which comes just after $2$ is $3$. Is there a real number which comes just after a particular real number? For example: Is there a number which comes just ...
0
votes
1answer
44 views

Which of two quantities is greater?

Let $x$ and $y$ be two positive real numbers such that $x>y$. Which of the quantities is bigger and when? $(x-y)\log\left(1-\frac{y}{x}\right)$ $x\log\left(1-\frac{y}{x+y}\right)$
3
votes
3answers
173 views

In-Depth Explanation of How to Do Mathematical Induction Over the Set $\mathbb{R}$ of All Real Numbers?

     I've seen in the answers to a few different questions here on the Mathematics Stack Exchange that one can clearly do mathematical induction over the set $\mathbb{R}$ of ...
-2
votes
1answer
26 views

For which real x does the following series converge [closed]

For which $x\in\mathbb R$ does the series $\Sigma\ x^{n!}$ converge?
0
votes
1answer
26 views

The terminology for particular subsets of the power set of R

The set $X = \{\{x\in\Bbb R\mid x<a\}\mid a\in\Bbb R\}$ , which is a subset of the power set of R. Is there a terminology for the set $X$? My intent is to search the related literature about the ...
0
votes
2answers
35 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
0
votes
0answers
27 views

On the rearrangement of an infinite series of real numbers. [duplicate]

A chapter of a text book ended with. If we rearrange infinitely terms of a series that converges only conditionally, we may get results that are far different from the original series ...
1
vote
5answers
51 views

How is “smaller than” defined on $\mathbb{R}$?

According to http://en.wikipedia.org/wiki/Binary_relation Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in ...
2
votes
1answer
75 views

An awkward Functional Equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x+1))^2$$ for all $x,y \in \mathbb{R}.$ I proved that $f$ is bijective, but I am stuck there. Any help please?
1
vote
1answer
33 views

Closeness of infinite union of closed sets

Is the set $\bigcup_{x \geq 0} \left\{\frac{1}{x+1} \right\} $ closed? For all $x \geq 0$, the set $\left\{\frac{1}{x+1}\right\}$ is a single point, therefore it is closed. But I am not sure about ...
1
vote
0answers
47 views

Partitioning real numbers with sum $1$ to sets

If the sum of a finite number of positive real numbers is $1$ and each of them is less than $x$, then those real numbers can be partitioned into $50$ sets (some of which may be empty) such that the ...