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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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Rudin Theorem 1.17

1.17 THEOREM Let $f: X \to [0, \infty)$ be measurable. There exist simple measurable functions $s_n$ on $X$ such that (a) $0 \le s_1 \le s_2 \le \dots \le f.$ (b) $s_n(x) \to f(x)$ as $n \to \infty \forall x$ PROOF Put $\delta_n = 2^{-n}$. To each…
shimee
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If $\alpha,\beta,\gamma$ are the roots of equation $x^3-7x^2+12x-11=0$, find the last four digits of $\alpha^{21007}+\beta^{21007}+\gamma^{21007}$

If $\alpha,\beta,\gamma$ are roots of the equation $x^3-7x^2+12x-11=0$, then find the last four digits of $\alpha^{21007}+\beta^{21007}+\gamma^{21007}$. I know that $\alpha+\beta+\gamma=7,…
vqw7Ad
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How is the boundary of the clopen set [0,1) empty?

I don't get why the boundary of a clopen set is empty. If you take $A = [0,1)$ in $\mathbb R$, then isn't the closure of this the smallest closed super set that contains $A$ which is $[0,1]$. Isn't the interior, $(0,1)$? So the boundary would be…
Andy
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Showing uniqueness of Riemann's Integral

I am given the definition: Le $f$ be defined on $[a,b]$. we say that $f$ is Riemann Integrable on $[a,b]$ if there is a number $L$ with the following property: for every $\epsilon>0$, there is a $\delta > 0$ such that $\left\|P\right\|< \delta$…
user70337
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Showing $x_{n}=x_{n+1}$ for $n\geq k_{0}$.

Let $\{x_n\}_{n\geq 1}$ be a sequence of positive integers such that the there is a $k\in\mathbb{N}$ such that the sum $$\frac{x_1}{x_2}+\frac{x_2}{x_{3}}+\cdots + \frac{x_n}{x_1}\in\mathbb{N}$$ for all $n\geq k$. I have to show that there is an…
Ramana
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Absolutely continuous functions and the fundamental theorem of calculus

Wikipedia says that the function $$\begin{equation} f(x) = \begin{cases} 0 & \mbox{if } x = 0 \\ x\sin(1/x) & \mbox{if } x \neq 0 \end{cases} \end{equation} $$ is not absolutely continuous on any finite interval containing the origin.…
user782220
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What does the Heine-Borel Theorem mean?

Question: What is the Heine-Borel Theorem saying? I am working through Hardy's Course of Pure Mathematics (ed.3) and am attempting to understand what the Heine-Borel means (s. V, pp. 186). Theorem: Suppose that we are given an interval $(a,b)$ and…
GovEcon
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Proving $K$ is compact directly.

If $K$ is a subset of metric space $\mathbb{R}^n$ and if every real valued continuous function on $K$ is bounded, then $K$ is compact. I know a proof considering $K$ is unbounded and not closed. This is proof by contradiction. Is there any…
Learning
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Help with Baby Rudin Theorem 5.5 Proof

I have a question about the last sentence. I know $s \rightarrow y$ as $t \rightarrow x$; hence $u(t) \rightarrow 0$ as $t \rightarrow x$. But I'm confused that why the term $v(s)$ disappears. That is, why $v(s) \rightarrow 0$ as $t\rightarrow x$?…
Xiangdong Meng
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Let $h:[0,1] \times [0,1] \rightarrow \mathbb{R}$ be the function $h(x,y)=f(x)g(y)$. Show h is integrable.

Let $f,g:[0,1] \rightarrow R$ be bounded, nonnegative, and nondecreasing $f(x_1) \leq f(x_2)$ for all $x_1 \leq x_2$ functions. Let $h:[0,1] \times [0,1] \rightarrow \mathbb{R}$ be the function $h(x,y)=f(x)g(y)$. Show h is integrable. Theorem: Let…
neveryield
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Gabriel's Horn Painter's Paradox

How will you explain the painter's paradox in the Gabriel's horn that even if we require infinite amount of paint to cover the surface of the Gabriel's horn but finite amount of paint is sufficient to fill it with paint?
Pratik Rawal
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Properties of min(x,y) and max(x,y) operators

Is $\min(x^2,y^2)=[\min(x,y)]^2$, and similarly for $\max(x,y)$? Also, is $\sqrt{\min(x^2,y^2)}=\min(x,y)$? Do other non-linear operations work? In general, what are the other interesting properties of these operators, and where can I study more…
PGupta
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Interpretation of zero angle between two elements in a inner product space

Take $f,g \in V$, where $V$ is an inner product space. Let $\langle \cdot, \cdot \rangle : V \times V \to [0,\infty)$ denote the inner product operator in $V$. Let the "angle" $\theta$ between $f$ and $g$ be defined through the rule $$ \cos(\theta)…
Ulises Nunez
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Real positive number less than one as a series of rationals

Let $0
user2820579
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Twice continuously differentiable bounded functions with non negative second derivative

Let $f: \mathbb R \rightarrow\mathbb R $ be twice continuously differentiable. Suppose further that $f$ is bounded and $f’’(x)\geq 0$ for every $x \in \mathbb R $. Then prove that $f$ is infinitely differentiable. Except constant functions I am not…
Lawrence Mano
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