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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

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Convergence of ess-sup w.r.t. a sequence of empirical measures

Given a Polish space $E$ endowed with Borel $\sigma$-algebra. We consider a sequence of empirical measures $(\eta_N)_{N> 0}$ such that for all the bounded measurable test functions $f\in\mathcal{B}...
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The continuous functions of compact support are dense in $L^{1}$.

Theorem The following family functions are dense in $L^{1}(\mathbb{R}^{d})$: (i) The simple functions (ii) The step functions (iii) The continuous functions of compact support. Proof: ...
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A specific problem on : Does bounding of the Sobolev norm can cause bounding of a higher derivative?

Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
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Elementary(?) measure theory.

A student complained that he was stuck on an exercise: Suppose $(f_n)$ is a sequence of measurable functions on some measurable space. The set of $x$ such that $\lim_nf_n(x)$ exists is ...
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An Axiomatization of Magnitudes Using Only Three Primitive Notions (c.f. Tarskis' Real Numbers)

For background, see Tarski's axiomatization of the reals. Here we are seeking axioms to categorize the positive real numbers. The Primitive Notions: A constant $1$. A set of magnitude numbers $M$. ...
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25 views

Measure on ([0,1], P([0,1]))

Is there any measure on $([0,1], P([0,1]))$ which satisfies: $$ \mu ([0,1]) = 1 $$ $$ \mu(A) \in \{0,1\} $$ for any $A \subset [0,1]$ $$\mu(B) = 0$$ for any $ B \subset [0,1]$ finite. My "guess" is ...
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Can we apply L'Hopital's rule with respect paremeter $u$?

If we know that $\int_1^\infty g(x,t_o)dx=0$ and $\int_1^\infty h(x,t_o)dx=0$ $ t $ is real variable here $$f(x,t_o)=\lim_{t \rightarrow t_o} \frac{\int_1^\infty g(x,t)dx}{\int_1^\infty h(x,t)dx}= ...
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2answers
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Convergence of a constant sequence in a set of finite elements

In Munkres Topology section 16 in the subsection on Hausdorff Spaces there is a motivating example involving the three-point set $\{a,b,c\}$ which states that the sequence defined by setting $x_n=b$ ...
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1answer
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real analysis - subsequence of bounded sequence converges to $\lim_{k\to\infty}$ sup $a_k$

My professor proved that a subsequence of a bounded sequence $(a_k)$$_k$$_\to$$_\infty$ in $\mathbb R$ converges to $\limsup_{k\to\infty}$ $a_k$ this way: Denote a* = $\limsup_{k\to\infty}$ $a_k$ ...
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Definition of Hardy-Littlewood maximal operator using open balls or closed balls

Hardy-Littlewood maximal operator is defined by $$ Mf(x):= \sup_{B\ni x} \frac{1}{|B|} \int_{B} |f(y)| dy. $$ Here, the supremum is taken over balls $B$ in $\mathbb{R}^n$ which contain the point $x$ ...
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Interpolating a number of equidistant points.

I have the coordinates: $(0,3125),(1,3125), (2,2500), (3,1500), (4,600), (5,120), (6,0), (7,0), (8,0), (9,0), ...$ And I want a way to construct a smooth curve through the points that increases in ...
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2answers
41 views

real analysis - bounded sequence and lim sup

The question is Let $(a_k)_k$ $_\in$ $_\mathbb N$ be a bounded sequence of real numbers. Let b be a real number such that b > lim sup$_k$$_\to$$_\infty$ $a_k$ Prove that there exists N $\in$ $\...
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1answer
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construction of a step function which approximates the characteristic function

Let $S=\mathbb{Q}\cap [0,1]$. Let $1_S$ be the characteristic function regarding S. I want to show that for all $\epsilon >0$ there exists a step function $f$ satisfying $\| f-1_S\|_1 <\epsilon$...
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23 views

Differentiate under consideration of the area [on hold]

How can I calculate the following term: $\frac{d^2}{dt^2}\int_{\Omega_t}dx$ with $\Omega_t = T(\Omega)$, $T\in C^\infty$ and $\Omega$ is of class $C^3$. I do not want to change my area.
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1answer
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Difficulties understanding a solution of a real analysis problem

I was working on some problems on Integration part of the book Problems in Mathematical Analysis by Kaczor and Nowak. And it seems that I don't get this one solution. The sentences underlined with ...
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28 views

if $\int_{\mathbb R} f(x) dx \cdot v < \infty$, is it possible that $\int_{\mathbb R} f(x)\cdot v\ dx $ diverges?

Placing a dot product inside the vector valued integral can "tame" bad integrands. A toy example is if $\operatorname{bad}(x)$ is some bad integrand, maybe like $-x^{100} + \frac{1}{x^{100}}$, then ...
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1answer
49 views

Proof that $x^3$ is continuous everywhere on $\mathbb{R}$

If we have $$|x-x_0| < \delta$$ Then, $$||x| - |x_0|| < |x-x_0| < \delta$$ $$|x| - |x_0| < \delta$$ $$|x| < \delta + |x_0|$$ We need $$|x^3-x_0^3| < \epsilon$$ Thus, $$|x^3-x_0^...
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a quadratic functional that is coercive, has a unique minimum.

Is there a theorem that says, "a quadratic functional that is coercive, has a unique minimum". Appreciate any references.
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Limit of derivative of moment generating function

Suppose $X>0$ is an integer-valued with $P(X=k)=q_ke^{-t_0k}$, where $t_0>0$ and $\{q_k\}$ satisfies $\frac{1}{k}\log q_k \rightarrow 0$. Let $\phi(t)$ be the MGF of $X$. Let $t_{\text{max}}=\...
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1answer
29 views

Does a real number with this decimal expansion for $r$ and $r^2$ exist?

Does there exist a real number $0< x <1$, such that the decimal expansions of $x$ and $x^2$ are the same, starting from the millionth term, and neither expansion has an infinite ...
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1answer
46 views

Interior Points Question

Solution Attempt As $A \subseteq A \cup B$ we have $Int(A) \subseteq A \subseteq A \cup B$ Similarly, $Int(B) \subseteq A \cup B$ $\implies Int(A) \cup int(B) \subseteq A \cup B$ Now $Int(A) \cup ...
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42 views

Compute $\lim \limits_{n\to \infty} n^{1/n}$

Two part question that I want to make sure I did correctly. a) Let $x_n = \sqrt[n]{n} - 1$. Use the fact that $(1 + x_n)^n = n$ to show that $x_n^2 \leq \frac{2}{n}$. Hint given to use the binomial ...
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0answers
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cost of gaussian elimination in numerical

In the following question I have solved the system $$2x_1-x_2=2$$ $$-x_1+3x_2 = 4$$ Using gaussian elimination. Below are the workings, however I am wonder what is the cost of this process? how do I ...
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1answer
22 views

Lebesgue Integration in practice on a bounded continuous function.

Suppose a continuos lebesgue measuable function $f$ that is (non negative) bounded above by $M$. Define a sequence (similiar to standard represtation in counting measure) $\displaystyle f_m=\sum_{i=0}...
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Matching intuition of convergence with formalism for family of spherical caps

Consider the following family of surfaces in $\mathbb R^3$. Let $P=(0,0,h)$ with $h>0$. Construct the sphere $S$ of radius $R$ centered at $(0,0,-R)$. Now draw the cone with vertex at $P$ that is ...
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1answer
33 views

For which values of $a$ does the series converge?

For which values of $a$ does the series below converge? $$\sum_{n=1}^∞ \frac{(\ln n)^{2014}}{n^a}.$$ The answer is $a > 1$. I do not have any idea how to do it. I have tried the ratio test and the ...
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Proof of Generalised ratio test

Is the following argument correct? Proposition. Suppose $\{x_n\}$ is a sequence and suppose for some $x\in\mathbf{R}$, the limit $$L:=\lim_{n\to\infty}\frac{|x_{n+1}-x|}{|x_n-x|}$$ exists and $L<1$...
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2answers
70 views

$X(\omega)=\sum_{n=1}^{\infty} X_n(\omega)2^{-n}$ is a Uniform Random Variable?

I am working on a problem on the infinite coin-tossing space and I'm having trouble making any meaningful progress. Let $(\Omega, \mathcal F, P)$, where $\Omega=\{0,1\}^{\mathbb N}$, $F$ is the $\...
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Examples where we cannot interchange summation and integration

We know of conditions under which we can interchange summation and integration (1, 2). What are some simple examples where we cannot do so and which we could present to high-school/introductory ...
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3answers
144 views

Bound on $\sum\limits_{n=0}^{x}{\sin{\sqrt{n}}}$

Using Desmos and Mathematica, I was able to find a function $g(x)$ that seemingly estimated the function $$f(x)=\sum_{n=0}^{x}{\sin{\sqrt{n}}}$$ I found that $${g(x)=2\sqrt{x}*\sin{\left({{\sqrt{...
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2answers
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Equality of simple functions with measures.

Let $(X,\mathcal{M},\mu)$ be a measure space. Let $a_i, b_j \geq 0$ and $A_i, B_j \in \mathcal{M}$ of finite measure, for $1 \leq i\leq n$, $1 \leq j \leq m$. If $\sum_{i=1}^n a_i\chi_{A_i} \...
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Find the intersection of two functions, $f(x) = \tan x$ and $g(x) = 5x$.

I can't proceed with the normal method to find these ones because they will intersect infinite times. Should I take an interval? $$f(x)=\tan(x), \qquad g(x)=5x$$ Thank you.
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2answers
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Determine the number of point $z \in \mathbb{C}$ such that $(2z+i\overline{z})^3=27i$

I've just learned complex numbers in Mathematical Analysis 1, and I'm stuck in the following problem: I would like to determine the number of point $z \in \mathbb{C}$ such that $(2z+i\overline{z})^3=...
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2answers
61 views

Evaluate $\lim_{x\to0}\frac{\sqrt{a^2+x}-|a|}{\sin^a2x\ln\cos x}$

I have just learned limits and series by myself, but I'm stuck with this limit: $$\lim_{x\to0}\frac{\sqrt{a^2+x}-|a|}{\sin^a2x\ln\cos x}$$ I would like to evaluate that limit with $a\in\mathbb R$. I ...
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1answer
21 views

Show that in any metric space, intersection of any number of compact sets is compact. [duplicate]

Show that in any metric space, intersection of any number of compact sets is compact.
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2answers
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One-sided continuity and derivative

If you are given $f(x):\Bbb{R} \rightarrow \Bbb{R}$ defined on $[c,\infty)$ and $\displaystyle\lim_{x \to c^+}{f(x) = f(c)}$ Then, we say that $f(x)$ is "right continuous" at $c$ Is it the case ...
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20 views

Continuity along all curves imples continuity everywhere

Let $f:\mathbb R^2\to\mathbb R^2$ be a function such that $f\circ \gamma$ is continuous for all curves $\gamma:[0,1]\to\mathbb R^2$. Is it necessary that $f$ is continuous?
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1answer
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Another way to prove $d(x,y)$ is a metric

Today on an exam, I was tasked with proving that for all $x,y$ in $R_+^n$, $d(x,y)=\sqrt{\sum_{i=1}^n(\sqrt{x_i}-\sqrt{y_i})^2 }$ is a metric. It's pretty clear that $d(x,y)=0$ iff $x=y$, and also ...
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1answer
47 views

Help with past Analysis prelim problem.

Let $f$ be a nonnegative Lebesgue measurable function on $[0,\infty)$ such that $\int_0^{\infty}f(x)dx < \infty$. Show that there exists a positive, non-decreasing, Lebesgue measurable ...
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1answer
30 views

radially unbounded functions

Is the following function radially unbounded or not? $$V(x) = \frac{x_{1}^2}{1 + x_{1}^2} + x_{2}^2$$ I know that if $x_{2} \to \infty$ in which case $||x|| \to \infty$ and $V(x) \to \infty$ but if $...
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0answers
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Regulated Indicator Function [on hold]

$$ \text { Let } \left\{ q _ { 1 } , q _ { 2 } , \ldots \right\} \text { be an enumeration of the rationals in } [ 0,1 ] , \text { and define } $$ $$ f _ { n } ( x ) = \left\{ \begin{array} { l l } { ...
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0answers
37 views

Why do continuous partials imply differentiability?

Wikipedia defines differentiability of a multivariate function as follows: A function of several real variables $f: \Bbb{R}^m \rightarrow \Bbb{R}^n$ is said to be differentiable at a point $x_0$ if ...
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1answer
21 views

Proof Cauchy-Schwarz inequality

I want to proof $|\langle x, y \rangle| \leq \|x\| \, \|y\|$ for all $x,y \in \mathbb{R}^n$ or $\mathbb{C}^n $. I know there exist a ton of proofs for this inequality, but it want to proof it through ...
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1answer
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An embedding of the line in a plane.

Prove that there is an embedding of the line as a closed subset of the plane, and there is an embedding of the line as a bounded subset of the plane, but there is no embedding of the line as a closed ...
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0answers
17 views

Pointwise convergence of monotonically increasing functions [duplicate]

Let $\left( f _ { n } \right)$ be a sequence of functions from $[ a , b ]$ into $\mathbb { R } .$ Prove that if $\left( f _ { n } \right)$ converges pointwise on $[ a , b ]$ to a continuous function $...
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3answers
42 views

Compute $\frac{d}{dx}\int_a^{x^2}f(t)dt$ for continuous $f$

Question: If $f$ is continuous on $[a,b]$, then compute $\frac{d}{dx}\int_a^{x^2}f(t)dt$. My attempt (may be wrong ): let $F$ be the antiderivative of $f$. Then $\frac{d}{dx}\int_a^{x^2}f(t)dt=\frac{...
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0answers
16 views

Convergence of Complex-valued functions [on hold]

Suppose that $(X,\mathscr{M},\mu)$ is a finite measure space, that is, $\mu(X)<\infty$, and that $f,f_1,f_2,...$ are complex-valued measurable functions on $X$. Prove that $f_n\to f$ $\mu$-a.e. if ...
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1answer
20 views

Show that a subset of a linearly ordered set might not have a least element.

I am quite confused about the definition of least element. I know this definition of least element though from Schramm's Real Analysis If S is a subset of an ordered set, a least element of S is an ...
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2answers
28 views

Proving a contrapositive of the ratio test for sequences.

Let $(a_n)$ be a sequence s.t. $a_n\rightarrow L\in\mathbb{R}\setminus{\{0\}}$, let $\forall n\in\mathbb{N},a_n>0$. We wish to show that $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=1$. (From definition). ...
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0answers
37 views

Show that $| u_2 + v_2 -1 | - |u_1 + v_2 -1| - |u_2 + v_1 -1| + |u_1 + v_1 - 1| \geq 0$

Quick question, assume $u_1 \leq u_2$, $v_1 \leq v_2$ $\in [0,1]$. I'm trying to show that $$ | u_2 + v_2 -1 | - |u_1 + v_2 -1| - |u_2 + v_1 -1| + |u_1 + v_1 - 1| \geq 0$$ My idea is that the ...