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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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Continuity of the map $f:0.a_1a_2a_3\ldots \mapsto 0.0a_10a_20a_3\ldots$

I'll first state the question: Let $f:[0,1] \to [0,1]$ be a function defined as follows: $f(1)=1$, and if $a=0.a_1a_2a_3\ldots$ is the decimal representation of a (which does not end with a chain of 9's), then $f(a)=0.0a_10a_20a_3\ldots$ . Discuss…
Sayantan
  • 3,418
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A continuous function on a sphere attains the same value at some pair of antipodal points

Let $S=\{x\in \mathbb R^n: \|x\|=1\}$ be the unit sphere in $\mathbb R^n$, and let $f: S\to \mathbb R$ be a real-valued continuous function on $S$. Prove that there is a point a belonging to $S$ such that $f(a)=f(-a)$.
Cory
  • 91
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Riemann Mapping Theorem for Homeomorphisms

We know by Riemann mapping theorem that for every connected, simply connected proper open subset $U$ of the complex plane, there exists a biholomorphic map $\phi:U\to \mathbb{D}$ (Where by $\mathbb{D}$ we mean the unit open disk). The proof of this…
Hesam
  • 735
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Proof of limit of sum of series

I know that $$\exp(-x)=\sum_{n=0}^\infty\frac{(-1)^{n+2}x^n}{n!}$$ converges for every positive $x$ Then we should have $$\lim_{x\rightarrow+\infty}\sum_{n=0}^\infty\frac{(-1)^{n+2}x^n}{n!}=0$$ How can we prove the above result WITHOUT using the…
anonymous67
  • 3,458
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2 answers

Archimedean Principle

Prove the Archimedean principle for $\mathbb{Q}$, the set of rational numbers. I know the proof for real numbers If not, then the nonempty set $S = \{nx \mid n ∈ \mathbb{N}\}$ has an upper bound $y$, and so by the least upper bound property of the…
Amir Suhail
  • 41
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Proving "No matter how large a real number $x$ is given, there is always a natural number $n$ larger"

I know the statement above is true because of the Archimedian Theory. Would the following proof make sense to prove it? This is a proof by contradiction. If the set of natural numbers does have an upper bound, then it has a least upper bound. Let $x…
knerd
  • 83
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Stone-Weierstrass theorem proof in Rudin

I'm reviewing Rudin's Principles of analysis now. He first proves Weierstrass approximation and uses it to generalize to Stone-Weierstrass theorem. The problem is, this version is very restrictive. What he proves here is that: Let $X$ be a compact…
Rubertos
  • 12,491
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Prove $\exists$ closed subset A st $f(A)=A$

Let $(X,d)$ be a compact metric space and $f$ be continuous on $X$, show that there exists non-empty closed subset $A$ of $X$ such that $f(A)=A$ So, $f$ will be uniform continuous, but how does that help?
Mathronaut
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Supremum of a Set of Integers

I am working on the following problem: If a set of integers $S \subset \mathbb{N}$ has a supremum, show that $\sup S \in S$. My approach is as follows: Let $s_0 = \sup S$ and suppose $s_0 \notin S$. It is a fact that for every $\epsilon > 0$ there…
ItsNotObvious
  • 10,883
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About the domain of points having tangent to a curve

Let the graph of $y=f(x)$ be a curve $C$ and $f''>0$. Prove that if $y_0\leq f(x_0)$ then there exist a tangent of $C$ go through $(x_0,y_0)$ I don't know how to prove the existence.
anonymous67
  • 3,458
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Continuity of Thomae's Function at Irrationals

Thomae's Function, defined by $$T(x) = \begin{cases} \dfrac{1}{q} & \text{$x = \dfrac{p}{q}$, where $p,q\in\mathbb{Z}$ and $\gcd(p,q) = 1$, and} \\[1ex] 0 & \text{otherwise}. \end{cases} $$ is known to be continuous at irrationals and discontinuous…
Groups
  • 10,238
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Supremum of a subset is less or equal than infimum of another subset

Let $X,Y$ be two bounded subsets of $\mathbb{R}$ satisfying the following proposition 1 : $$\forall x \in X, \forall y \in Y : x \leq y $$ I wanted to know if there's a direct proof of $$\sup X \leq \inf Y$$ I think I managed to prove it by…
nerdy
  • 3,288
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Unit disc contains finitely many dyadic squares whose total area exceeds $\pi - \epsilon$

This is 1.22a in Pugh's Real Mathematical Analysis (p. 44): Given $\epsilon > 0$, show that the unit disc contains finitely many dyadic squares whose total area exceeds $\pi - \epsilon$, and which intersect each other only along their…
dmk
  • 2,228
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Showing $f\in L^\infty$ in a finite measure space

I am studying for a qualifying exam, and I seem to have difficulty working problems involving $L^p$-spaces. An explanation for the following problem would be very helpful! Let $(X, \Sigma, \mu)$ be a finite measure space and let $f$ be a real-valued…
phenomenalwoman4
  • 805
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Proof of rational density using Dedekind cuts

The Problem Let $x$ and $y$ be real numbers such that $y
Matt Rest
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