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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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Limsup inequality - how do I write this as a rigorous proof?

Problem: 1. Let $(a_n)$ and $(b_n)$ be bounded sequences of real numbers. Prove that $$\limsup _{n\to \infty}(a_n+b_n)\leq \limsup _{n\to \infty}a_n + \limsup _{n\to \infty}b_n,$$ and that equality holds when $(a_n)$ or $(b_n)$ is convergent. Weak…
JohanLiebert
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Limit of integrals around the origin

Let $1
Johannes
  • 85
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Proving an inequality:

I have this inequality but I am unsure how to prove it: 0$<\alpha \leq$1 a$^\alpha$+b$^\alpha$ $\geq$ (a+b)$^\alpha$ $\forall a,b \geq 0$ I was given a hint: we can assume b$\geq$0 $(\frac{a}{b})^\alpha$+1 $\geq$ $(\frac{a}{c}+1)^\alpha$ so it…
user99262
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$f: [0,1]\rightarrow \mathbb{R}$ be an injective function, then :

Question is : $f: [0,1]\rightarrow \mathbb{R}$ be a one one function, then which of the following statements are true? $(a)$ $f$ must be onto $(b)$ range of $f$ contains a rational number $(c)$ range of $f$ contains an irrational number $(d)$ range…
user87543
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extension of a function to a differentiable function

Suppose we have a map $f: S \rightarrow \mathbb{R}^{n}$, where $S \subset \mathbb{R}^{m}$, such that for each $a \in S$ there exists an $m$ by $n$ matrix $A$ such that $\lim_{h \rightarrow 0}\frac{f(a+h)-f(a)-Ah}{|h|} = 0.$ What conditions must be…
user13255
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1 answer

existence of equal function values for a continuous function

I am reviewing for a midterm and this a problem from a previous year's final. Assume that $f \in C ([0,2])$ and $f (0) = f (2)$ Prove that there exist $x_1$ and $x_2$ in $[0,2]$ such that $x_2 -x_1 = 1$ and $f (x_2) = f (x_1)$ I am not even sure…
user97549
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Prove that $f$ is continuous if and only if $\text{osc}(f,a) = 0$

Let $(X,d)$ be a metric space, let $f:X\rightarrow \mathbb{R}$, and let $a\in X$. Define the oscillation of $f$ at $a$ by $$\text{osc}(f,a) = \inf_{r>0}(\sup \{|f(x) - f(y)| : x,y \in B_r(a)\}).$$ (Note: $B_r(a) = \{x: |x - a| < r\}).$ Prove that…
JohanLiebert
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If the limit exists the set is closed

Prove that if $p_n \to p$ in a metric space then the set of points $\{p,p_1,p_2, ...,\}$ are closed. A theorem in my book states that a set $S$ in a metric space is closed if and only if whenever $q_1,q_2,...$ is a sequence of points in $S$ that…
Tom
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Show that the sequence $1+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}$ is Cauchy

Show that the sequence $1+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}$ is Cauchy. I'm not sure where to start with this problem, I know that if I can show that the sequence is convergent I can manipulate the inequality definition of the limit to…
Duiliath
  • 85
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Construction of a certain function.

Sorry for the vague title but here is the question. Let $F := C_b([0,1],\mathbb R)$ and $$ U_n := \left \{f \in F: \forall x \in [0,1] \exists y \in [0,1]: \left \lvert \frac{f(x)-f(y)}{x-y} \right \rvert > n \right \} $$ I want to prove that…
user42761
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3 answers

absolute convergence implies unconditional convergence

Why is it true that absolute convergence implies unconditional convergence ? This is interesting, because the sequences (corresponding to each rearrangement) of partial sums are different, still they converge to the same value if the series is…
user21982
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1 answer

How can I explain this formally?

Let $f: \Bbb R \to \Bbb R$ and $x \in \Bbb R$. Suppose that $\lim_{y \to x+} f(y)$ exists as a real number. If there is an $r \in \Bbb R$ such that $$\lim_{y \to x+} f(y) > r$$ then there exists $n \in \Bbb N$ (dependind on $x$ and $r$) such that…
user94756
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2 answers

Supremum and infimum for $n^{1/n}$

I have the set $A=\{\sqrt[n]{n}: n \in \mathbb{N}\}$. I can see that $\inf A$ is $1$ as $\sqrt[1]{1}=1$ but I am having trouble with $\sup A$. I know that it is $\sqrt[3]{3}$ because it only decreases from then onwards. But I have trouble proving…
nanki
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Intro to proof in real analysis 1

This is what I have to prove: For elements $x, y$ in an ordered field, if $0 < x < y$ then $y^{-1} < x^{-1}.$ My proof: $0 < x < y$ multiply $x^{-1}$ on the left of both sides to get $x^{-1}0 < x^{-1}x < x^{-1}y $ …
user87274
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0 answers

Does Weak $L^{2}$ Convergence on Finite Measure Spaces Imply Strong $L^{1}$ Convergence?

When $\mu(X)<\infty$, $L^{1}(\mu)\supset L^{2}(\mu)$ and by the Riesz-Fischer theorem, weak convergence of $f_{n}\to f$ in $L^{2}$ is equivalent to $\int_{X}f_{n}\bar{g}\;d\mu\to\int_{X}f\bar{g}\;d\mu$ for each $g\in L^{2}(\mu).$ Taking…
Sargera
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