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Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
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Prove that $x,y$ are positive real, $x/y$ is irrational, then the set $\{ mx+ny:m,n\in \Bbb Z \}$ .is dense in $\Bbb R$

Anyone can help with the following problem, Prove that if $x,y$ are positive real, $x/y$ is irrational, then the set $\{ mx+ny:m,n\in \Bbb Z \}$ is dense in $\Bbb R$. Thanks a lot!
Tuyet
  • 847
6
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2 answers

Construct series

Are there two non-negative, monotone sequences ${\{a_n\}}$ and ${\{b_n\}}$, s.t. $\sum{a_n}$ and $\sum{b_n}$ diverge, but $\sum{min{(a_n,b_n)}}$ converges? I guess that the convergence speed must be carefully controlled, but I couldn't find such…
Tundoku
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How can I prove that this infinite product equals $0$?

I am having trouble to prove that $$\prod_{n=1}^\infty\frac{2n-1}{2n}=0$$ I know that the sequence of partial products $(p_n)$ converges to a limit $L\ge0$, because $$p_n=\frac{(2n)!}{2^{2n}(n!)^2}$$ is decreasing and bounded below by zero. I…
Fábio Xavier
  • 63
6
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4 answers

find $\lim\limits_{n \rightarrow \infty }(n+1)\int\limits_{0}^{1} x^n f(x)$

Let $f(x)$ , $x \in [0,1]$,be any positive real valued continuous function. Then find $$\lim\limits_{n \rightarrow \infty }(n+1)\int\limits_{0}^{1} x^n f(x)$$ I tried by integrating by parts to get the result $f(1)-(n+1)\int\limits_{0}^{1}…
user321656
6
votes
1 answer

Polygonally connected open sets

I cannot understand the following theorem: An open set $S$ in $\Re^n$ is connected if and only if it is polygonally connected. I would be thankful if some one could present an intuitive proof of this theorem. Thanks for reading!
Pedro Gomes
  • 3,891
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1 answer

Prove that $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k = \int \limits _E f$

Show that if $f : E \rightarrow [0,\infty]$, $\lim \limits _{k\rightarrow \infty} f_k = f$ on $E$, and $f_k \leq f$ on $E$ for each $k \in N$, then $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k = \int \limits _E f$ An idea was to show…
rioneye
  • 953
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When can the space under the graph of a function be filled with closed balls?

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function and denote the part of the plane under the graph of $f$ $$\mathcal{G} = \{(x,y)\in\mathbb{R}^2 \space : \space y\leq f(x)\}$$ The collection of all closed balls inside…
ploosu2
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$\lim_{r\to 0}\frac{\operatorname{vol}f(B(a;r))}{\operatorname{vol}B(a;r)}=|\det f'(a)|$

I'm trying to solve this question: Let $U\subset \mathbb R^m$ be an open set and $f:U\to \mathbb R^m$ a function of class $C^1$. Suppose there is $a\in U$ such that $f'(a):\mathbb R^m\to \mathbb R^m$ is an isomorphism. Show $$\lim_{r\to…
user42912
  • 23,582
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9 answers

There exists a rational sequence that converges to $\sqrt3$

I got a proof of this but I am quite sure that it is not what was expected on the exam. Also, this proof seems really kludgy and non-kosher. Because of the density of the raitonals in the reals, there exists a $q\in\mathbb{Q}$ such that…
fhyve
  • 2,495
6
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4 answers

Proof of continuity for a real function!

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a real function, and satisfy that: for all $x\in\mathbb{R}$ $$\lim_{r\to x,r\in\mathbb{Q}}f(r)=f(x).$$ Show that $f$ is continuous on $\mathbb{R}.$
Riemann
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2 answers

Absolute value of Lebesgue integrable function

I want to prove that a measurable function $f$ is Lebesgue integrable iff $|f|$ is. I've proved the first part but how can I show if $|f|$ is Lebesgue integrable then $f$ is ?
Bunny
  • 512
6
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1 answer

$\sin(1/x)$ not uniformly continuous

I looked at your answer to the question posted How to prove $\sin(1/x)$ is not uniformly continuous thank you for your helpful explanation on how to think about it. But I am failing to see why we will choose $x=\frac{1}{2πk−π/2}$ . I know that we…
user43901
  • 1,028
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2 answers

Is the sequence $\{\{\log(n!)\}\}_n$ dense in $[0,1]$?

I tried to find this problem in Mathematics Stack Exchange and in Math Overflow, but I didn't find it anywhere. Here is my problem: Is the sequence $\{\{\log(n!)\}\}_n$, the sequence of fractional parts of $\log(n!)$, dense in $[0,1]$? Note 1: Here…
Στέλιος
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Find the interior, accumulation points, closure, and boundary of the set

I need to find the interior, accumulation points, closure, and boundary of the set $$ A = \left\{ \frac1n + \frac1k \in \mathbb{R} \mid n,k \in \mathbb{N} \right\} $$ and use the information to determine whether the set is bounded, closed, or…
brdcastguy
  • 169
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3 answers

Prove that if $12a+6b+4c+3d=0$, then $a+bx+cx^2+dx^3=0$ has a real solution in $(0,1)$

Assume that $a,b,c,d$ are real numbers such that $12a+6b+4c+3d=0$. Prove that $a+bx+cx^2+dx^3=0$ has a real solution in $(0,1)$. Note : I have no idea ! How is the assumption even related to the statement that the question wants us to prove? I…
Tina Ashtiani
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