Mathematics Stack Exchange
Mathematics Stack Exchange

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
9
votes
1 answer

Uniform continuity and boundedness

I have come across a proof which I understand almost completely, except for one part: THEOREM: If $f$ is uniformly continuous on a bounded interval $I$, then $f$ is also bounded on $I$. PROOF: In this case we assume that $I$ is of the form $(a,b),…
Kristian
  • 1,245
9
votes
1 answer

Slight generalization of an exercise in (blue) Rudin

Exercise 7.20 of blue Rudin (Principles of Mathematical Analysis), 3rd edition, says: If $f$ is continuous on $[0,1]$ and if $$\int_0^1f(x)x^n\,dx = 0, (n=0,1,2,\ldots),$$ prove that $f(x)=0$ on $[0,1].$ One proof: Let $\{p_n\}$ be a sequence of…
Mike B
  • 411
9
votes
2 answers

Question 2.25 on Folland, Real Analysis

I'm trying to solve the exercises from Real Analysis - Moderns Techniques and Their applications by Folland and I have questios about the exercise 2.25 b). The statement: Let $f(x) = x^{-1/2}$ if $0
mathnewbie
  • 488
9
votes
2 answers

Field bigger than $\mathbb{R}$

Is there any field containing $\mathbb{R}$ for which every non-empty subset has an infimum and a supremum in that field? I am trying to understand whether $\overline{\mathbb{R}}$ (which is not a field) is the best possible.
jpv
  • 2,011
9
votes
2 answers

Is a function determined by its integrals over open sets?

If $f \in L^1(\mathbb R)$ satisfies $$ \int_U f = 0 $$ for every open set $U \subset \mathbb R$, then is it true that $f = 0$ a.e.?
analysis_qual
  • 189
9
votes
1 answer

Existence of $2^{1/2}$

I'm trying to prove, that the square root of 2 exists in $\mathbb{R}$. I'm looking at the set $A:=\{y\in\mathbb{R}: y\geq 0, y^2\geq 2\}$. I have already proven that the greatest lower bound $b$ of $A$ exists. However, now I don't know how to…
Russel
  • 91
9
votes
2 answers

Is there a sequence $(a_n)$ so that for every $r \in \mathbb{R}$, there is a subsequence of $(a_n)$ convergent to r?

My guess is that there is no such sequence and prove it by contradiction. My attempt is as follows: Let $A_r$ denote the set containing all terms of such subsequence for each $r \in \mathbb{R}$. Since each subsequence that converges to $r$ contains…
Shuyi Leo
  • 175
9
votes
5 answers

Construct a compact set of real numbers whose limit points form a countable set.

I searched and found out that the below is a compact set of real numbers whose limit points form a countable set. I know the set in real number is compact if and only if it is bounded and closed. It's obvious it is bounded since $\,d(1/4, q) < 1\,$…
InfimumMaximum
  • 1,406
9
votes
2 answers

Uniformly convergence in compact sets

(Theorem 7.13 in Baby Rudin) Suppose $K$ is compact, and (a) $\{f_n\}$ is a sequence of continuous functions on $K$, (b) $\{f_n\}$ converges pointwise to a continuous function $f$ on $K$, (c) $f_n(x) \ge f_{n+1}(x)$ for all $x \in K, n = 1, 2, 3,…
David Tan
  • 745
9
votes
1 answer

If $m^*(E)=\infty$, then $E=\bigcup_{k=1}^{\infty}E_k$, $E_k$ measurable and $m^*(E_k)<+\infty$

Reading Royden's fourth edition of Real Analysis. I'm working with outer measure defined as $$m^*(E)=\inf\left\{\sum_{n=1}^\infty l(I_n):\,E\subset \bigcup_{n=1}^\infty I_n\right\},$$ where each $I_n$ is a bounded, open interval. Also, $E$ is…
David
  • 2,262
9
votes
2 answers

The number of limit points of the set $\left\{\frac1p+\frac1q:p,q \in \Bbb N\right\}$ is which of the following:

I am stuck on the following problem: The number of limit points of the set $\left\{\frac1p+\frac1q:p,q \in \Bbb N\right\}$ is which of the following: $1$ $2$ Infinitely many Finitely many If I take $p$ to be fixed (say=$k$) and let $q \to…
learner
  • 6,726
9
votes
4 answers

Does the converse of uniform continuity -> Preservance of Cauchy sequences hold?

We know that if a function $f$ is uniformly continuous on an interval $I$ and $(x_n)$ is a Cauchy sequence in $I$, then $f(x_n)$ is a Cauchy sequence as well. Now, I would like to ask the following question: The function $g:(0,1) \rightarrow…
Sigma
  • 91
9
votes
5 answers

What's the rationale behind rejecting limits of recurrence relation?

I found this problem in Understanding Analysis by Stephen Abbott. Define a recurrence relation as $x_1=3$ and $x_{n+1}=\frac{1}{4-x_n}$. Find the limit of the sequence $(x_n)$. There are other parts to the question, but what I want to ask is: why…
koifish
  • 2,779
9
votes
4 answers

Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.

Suppose that $f(x)$ is continuous on $(0, +\infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function. My attempt is to show that for any point $a \neq b$ , we have $f(a)=f(b)$. But I have no idea on how to get this.…
Idonknow
  • 15,643
9
votes
1 answer

Distributional derivative coincides with classical derivative?

I am having some trouble understanding the precise meaning of the following statement: "if $f \in C^1 (\Omega)$ for some $\Omega \subset \mathbb R^n$, then the distributional derivative of $f$ coincides with its classical derivative". I know that…
Jose L. Lykón
  • 479
1 2 3
…
99 100