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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. .

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If $\lim_{n\to\infty}\int_0^1 f_n(x)dx=0$, are there points $x_0\in[0,1]$ such that $\lim_{n\to\infty}f_n(x_0)=0$?

This is part of an old qual problem at my school. Assume $\{f_n\}$ is a sequence of nonnegative continuous functions on $[0,1]$ such that $\lim_{n\to\infty}\int_0^1 f_n(x)dx=0$. Is it necessarily true that there are points $x_0\in[0,1]$ such that…
YN Chew
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a "tweak" to an infinite series - still convergent?

suppose that for a sequence of reals $(x_t)_{t\in\mathbb{N}}$ it holds that $\frac{1}{T}\sum_{t=1}^T x_t \rightarrow 0$, for $T\rightarrow \infty$. How do I show (sorry, this might be an embarrassing question) that $\frac{1}{T^2}\sum_{t=1}^T t x_t…
s_2
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Averages of Indicator Functions of Compact Sets in $\mathbb{R}^{2}$ Over Circles

This question comes from problem #3 of the Spring 2013 Analysis Qual Exam here http://www.math.ucla.edu/grad/handbook/hbquals.shtml Define for $f\in…
Sargera
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How does this proof of density of $\mathbb{Q}$ in $\mathbb{R}$ require $a \geq 0$?

My Real Analysis text offers the following proof of "given two real numbers a and b, with $a < b$, there exists a rational number r strictly in between of those two", conditional on $0 \leq a < b$. It also says that the case of $a < 0$ follows…
confused
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Prove that $x^{1/n} > x^{1/(n+1)}$ given that $x>1$.

Prove that, for any $x > 1$:$$x^{1/n} > x^{1/(n+1)}$$ It looks like induction to me but since there's no equality, I don't know what to substitute in anywhere.
Nick
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Function maximum with positive derivative

My text book contains the following task which I'm unsure of: Be $f: [a, b] \rightarrow \mathbb{R}$ differentiable in $b$ and $f\;'(b)>0$. Prove that $f$ contains an isolated local maximum at $b$ (this means there is a $\delta > 0$ with $f(b) >…
CuStud
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"Limit set" of function at point

Let $f \colon [a,b] \to \mathbb{R}$, and for $x \in [a,b]$, define $$L(x) = \{y \in \mathbb{R}: \text{there exists a sequence} \left(x_n\right) \text{with } x_n \to x \text{ and } f(x_n) \to y\}.$$ In other words, $L(x)$ is the set of all the…
Prasiortle
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If $ 2 a_n \leq a_{n-1} + a_{n+1} $ then $ a_{n+1} - a_{n} \to 0 $

Let $ a_n $ be a bounded sequence of real numbers such that $ 2 a_n \leq a_{n-1} + a_{n+1} $ for $ n = 2,3,...$ . Show that $ a_{n+1} - a_{n} \to 0 $
mr.stealyourgirl
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Continuity points of a inf fucntion

Let $a \geq 0$, and the application $T_a$ from $C(\mathbb{R}^+, \mathbb{R})$ with values in $\mathbb{R}^+\cup \{ \infty \}$ defined by $\inf \{ t \geq a , f(t) \geq a \}$. I for all $t \geq 0$, $f(t) < a$, then $T(f) = \infty$ I do not think this…
user1240705
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If $f(x) = 0$ has infinitely many solutions for Taylor series of $f$, does this uniquely determine $f$?

Euler's original proof that $\sum_n 1/n^2 = \pi^2/6$ seemed to implicitly rely on an assumption that the series $1 - x/3! + x^2/5! + \ldots$ was the only series that had $\{(n \pi)^2\}_{n=1}^{\infty}$ as the roots. Is this true in general? I.e. if…
user2566092
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Analysis Qualifying exam question UM May 2013

I am studying for qualifying exams and I found the following problem from University of Michigan May 2013 analysis qualifying exam. Let $f$ be a measurable function on $(0, \infty)$. Let $p > 1/2$ and define $g(x) = (x^p + x^{−p})f(x)$. Show…
Lucy Lewis
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Sharp constant $C$ for $\|x+y\|_p \leq C\| tx + (1-t)y\|_p$.

Let $1\leq p<\infty$ and we define $l_p$ space be the vectors $x= (x_1, x_2, \dots)$, such that $$\|x\|_p = \left(\sum_{n=1}^\infty \|x_n|^p\right)^{1/p}<\infty$$ Find the minimum constant $C$ such that for all $t\in [0,1]$ and vectors $x, y \in…
Hgtcl
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Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that $f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q}$

Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that $$f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q}$$ I found a solution here which is this: By translation we may assume $f(0)=0$. Fix $a\in {\bf Q}$ and…
zaemon_23
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Show that the norm of the maximum induces a Banach space

Let $\mathbf{P}$ The set of polynomials of degree $\leq$ two, with real coefficients in $[0, 1]$, if $l=ax^2+bx+c$ in $\mathbf{P}$, we define $||l||_{\infty}=\max_{x\in[0, 1]}{\{|l(x)|\}}$, i want to show thtat $(\mathbf{P}, ||l||_{\infty})$ is a…
Wrloord
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Two questions about a proof on this site: show $\displaystyle\lim_{n\to\infty} \int_{0}^{\infty} \frac{x^{1/n}}{(1+\frac{x}{n})^{n}} \,dx = 1$.

Please see this answer: https://math.stackexchange.com/a/2266612/51407 I want to know why $f_{n}(x)=\frac{x^{\frac{1}{n}}}{( 1+ \frac{x}{n})^{n}} \le \frac{1}{x^{2}}$? Also, the choice of the domaninating function $g$ is a bit confusing. I can see…
Mark
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