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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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Bounded and finite - examples of distinction

What is the distinction between a bounded function, and a finite function? Is there any example of two functions that satisfies only one of them? Definition If $|f(x)|<+\infty \forall x \in E$, we say $f$ is finite. Definition If there exist a…
1LiterTears
  • 4,572
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Limit point intuition

Quoting Rudin, "A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q\not=p : q \in E$." This would imply that the points in an open ball would all be limit points, since for any $p$ in $E$ there are $q$ such…
Don Larynx
  • 4,703
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Limit points of a sequence as limit points of subsequences

Let $\left(x_{k_n^1}\right)_{n \geq 0}$ , $\left(x_{{k_n^2}}\right)_{n \geq 0}$ , $\dots$, $\left(x_{{k_n^t}}\right)_{n \geq 0}$ (where $t \in \mathbb{N^*}$) subsets of set $\left(x_n\right)_{n \geq 0}$ such that the sets $\left({k_n^1}\right)_{n…
MathLearner
  • 750
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Show that $(\sqrt{x_n})$ is convergent with $|x_n - 4| < \frac{4n}{n^3+1}$ and $(x_n)$ is a sequence of nonnegative real number

Let $(x_n), n \ge 1,$ be a sequence nonnegative real numbers. If for any $n \ge 1$, $$|x_n - 4| < \frac{4n}{n^3+1},$$ then $(\sqrt{x_n})$ is convergent and find it's limit. Explain without any theorem. Attempt: First of all, I want to prove that…
math404
  • 445
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Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$ Prove that $S$ is dense in $\Bbb R.$

Let $$S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that $S$ is dense in $\Bbb R.$ My definition/knowledge of density is limited to only these two statements: A set $G$ is dense in $\Bbb R,$ iff for every $x,y\in \Bbb R$ , there exists a $g$ in…
Thomas Finley
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Null Sequences and Real Analysis

I came across the following problem during the course of my study of real analysis: Prove that $(x_n)$ is a null sequence iff $(x_{n}^{2})$ is null. For all $\epsilon>0$, $|x_{n}| \leq \epsilon$ for $n > N_1$. Let $N_2 =…
Damien
  • 4,291
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Prove there exists a real number $\alpha \in \mathbb{R}$ satisfying $\alpha^2 = 2$ by finding $\frac{1}{n_0} < \frac{\alpha^2 - 2}{2\alpha}$.

I'm currently self-studying from the book Understanding Analysis by Stephen Abbott. The book offers the proof that $\alpha^2 < 2$ cannot be the case and as an exercise asks to prove that $\alpha^2 > 2$ can also not be the case. The proof that…
alwayslearning
  • 43
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On the existence of a weight function making sequence of integration preserving limit

The problem goes as follows: Let $f_n$ be strictly positive Lebesgue measurable function defined on $[0,\infty)$ satisfying $$\lim_{n \to \infty} \int_0^\infty f_n(x)\ dx=0$$ then show that there exists a positive, strictly increasing measurable…
Roy Han
  • 911
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Bounded Sequences

I came across the following problems during the course of my self-study of real analysis: Show that the sequence $(x_n)$ defined by $x_n = 1+ \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ is unbounded. I know a sequence $(x_n)$ is bounded if…
Damien
  • 4,291
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Image of average convergent sequence

Consider sequence $\{ a_i \}_i$ such that $\lim_{n \to +\infty} \frac{1}{n}\sum_{i=1}^{n}a_i = A$. Assume that $f$ is a continous bounded function. Whether the sequence $\{ b_i \}_i$ with $b_i= \frac{1}{i}\sum_{j=1}^{i}f(a_j)$ is also convergent? I…
YCCCC
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Counterexample to converse of Weierstrass M-test

From Understanding Analysis by Abbott: Exercise 6.4.2 (c): If $\sum_{n=1}^{\infty} f_n$ converges uniformly on $A$, then there exist constants $M_n$ such that $|f_n(x)| \leq M_n$ for all $x \in A$ and $\sum_{n=1}^{\infty} M_n$ converges. My…
Arnold Davidson
  • 63
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A continuous representative of $L^2$-functions

Suppose I have a $L^2(\mathbb{R}^n)$ fucntion $f(x)$ such that $|f(x)-f(y)|
mnmn1993
  • 435
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Proving that $\lim\limits_{x\to 0}\frac{f(x)}{g(x)}=L$ implies $\lim\limits_{r\to 0}\frac {\int_R f(ry)h(y)\,dy}{\int_R g(ry)h(y)dy}=L$

Let $f$ and $g$ be functions defined in a neighborhood of $0$ in $\Bbb R$, such that $g(x)\neq 0$ in this neighborhood. Prove that for all $L\in[-\infty,+\infty]$ and for all non-negative functions $h$ with compact support such that $\int_{\Bbb…
user62138
  • 1,167
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Infimum of a set (II)

I came across another question about infimums of sets. Find $\inf \{(-1)^{n} + 1/n: n \in \mathbb{Z}^{+} \}$. Heuristaically, I think the infimum is $-1$ since, for large $n$, the second term goes to $0$ and the first term is at least $-1$. Let $A…
Damien
  • 4,291
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Is this identity true? $\lim\limits_{n \to \infty} x^{q_n r_n} = \lim\limits_{k \to \infty} (\lim\limits_{n \to \infty} x^{q_n r_k})$

I am working on a proof, which is complete, conditional on the following conjecture being true: Let $x$ be a positive real number; let $(q_n)_{n=1}^{\infty}$ and $(r_n)_{n=1}^{\infty}$ be convergent sequences of rational numbers (with the limits…
Jonas
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