Theoretical foundations of calculus: 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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6 views

Closed unit ball of complex functions

Suppose $B_1$ is the closed unit ball in $\mathcal{B}(E)=\{\text{bounded complex-valued functions on a set } E\}$, i.e. $B_1=\{f\in\mathcal{B}(E): d(f,f_0)=\sup\limits_{x\in E} \left| ...
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1answer
11 views

show $f\ast h$ continuous for $f\in C_c^\infty(\mathbb R^n)$, $h\in L^1(\mathbb R^n)\cap L^\infty(\mathbb R^n)$

I got the following exercise as homework and need a hint: Let $f\in C_c^\infty(\mathbb R^n)$ and $h\in L^1(\mathbb R^n)\cap L^\infty(\mathbb R^n)$. Show $f\ast h$ is continuous. My attempt: Let ...
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0answers
8 views

Lebesgue-Stieltjes Measure associated to $F$.

I would like some help here, please. First is confusing to me the definition of: Lebesgue-Stieltjes Measure associated to $F$. I'm reading Folland-Real Analysis, page 35, second paragraph. I do ...
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9 views

A question about n-dimensional Lesbegue measure

This is the p.70 of Folland Real analysis. It says that the completion of $m^n$ on $\mathcal{B}_{\mathbb{R}^n}$ is equal to the completion of $m^n$ on $\mathcal{L}^n$. But Why do they agree? Could ...
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4answers
38 views

Find $y^{(n)}(0)$ for every $n$.

Let $y(x)$ fulfill $y''-xy=0$. Furthermore: $y(0)=0,y'(0)=1$. Find $y^{(n)}(0)$ for every $n$. I tried different forms of recurrence relations but I couldn't do much with it without it becoming a ...
3
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1answer
29 views

Weak convergence - $f_n$ “goes up the spout”

Fix $1 < p < \infty$. Given $f \in L^p(\mathbb{R})$ define $f_n(x) = n^{1/p}f(nx)$ for $n = 1, 2, \dots$. Prove that $f_n$ converges weakly to $0$ in $L^p$. I'm really confised about this ...
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1answer
24 views

Confused about this definition of limit superior.

The definition I am given is as follows: Let $(x_n)$ be a real valued sequence. For each positive integer $n$, let $s_n:=\sup\{x_m:m\geq n\}$. If $(s_n)$ converges, we denote its limit by ...
0
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2answers
33 views

is this correct that if $\frac{\partial f}{\partial y}=0$ then $f$ is independent from $y$?

Suppose that $A=\{(x,y) \in \Bbb R^2 : x> 0 $ or $ y=0 \}$ and $f:A\to \Bbb R$ is an arbitary function. Prove that If $\frac{\partial f}{\partial x}=0$ then $f$ is independent from $x$ If ...
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2answers
20 views

Need a Probability Theory book that also focusses on Analysis

I am in search for a Probability Theory book which also contains elements and proofs from Analysis. A non-Measure Theoretic approach is most desirable. I have gone through great books like Ross but I ...
2
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1answer
31 views

What is the point of big Oh notation when it is used for estimation?

I'm reading a book on number theory at the moment that assumes familiarity with big Oh notation...and while I think I do understand the notation I cannot understand the point of it. For instance let ...
2
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0answers
9 views

Limsup question $ \sum_{n=1}^N \sum_{m=1}^N x_{m-n} \leq N^2 \bigg( \limsup_{n \to \infty} x_n + o(1) \bigg) $

Reading notes on the Poincare Recurrence Theorem and I am a bit stuck with Theorem 1. For a measure preserving dynamical system $T: X \to X$ we have $\mu(T^{-1}E) = \mu(E)$. Why does it "easily ...
2
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1answer
44 views

Can this special case happen when working with L'Hopitals rule?

I am using this version of L'Hopital's rule Assume that $\lim_{x \rightarrow a}f(x)=\lim_{x \rightarrow a}g(x)=0$, and that the limit-value $\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}$ exists (could ...
2
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4answers
57 views

Find $\lim_{x\to \infty}{[({1\over e}(1+{1\over x})^x)]^x}$.

Find $\lim_{x\to \infty}{[({1\over e}(1+{1\over x})^x)]^x}$. I have been trying for hours using the continuity of $e$ and using L'Hopital rule but it gets really scattered and ugly. I am in despaire. ...
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0answers
10 views

A real valued function having IVP

Given $f:\mathbb R\rightarrow \mathbb R$ be a function which maps intervals to intervals. Suppose for each sequence $x_n\rightarrow x \exists M $ such that $|f(x)-f(x_n)|\leq ...
2
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0answers
16 views

Monotone Convergence Theorem (for real sequences) equivalent to the Least Upper Bound Property?

Some days ago I have asked this question to which André Nicolas gave a link to this paper which contained a proof of the Least Upper Bound Axiom from Monotone Convergence Theorem via Archimedian ...
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0answers
16 views

Does the $C^1$ norm for the boundary of a ball depend on the radius?

Does the $C^1$ norm for the boundary of a ball with radius $r$ depends on the radius $r$? I use the classical definition of the $C^k$ boundary of a bounded domain: $\Omega$ is a bounded domain in ...
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0answers
6 views

Bound all $k$-th derivatives by directional derivatives of order $k$

Assume $f\in C^k(\mathbb{R}^n)$, $x\in\mathbb{R}^n$, and $|(\partial_\xi)^kf(x)|\leq 1$ for all $\|\xi\|=1$. Which bounds do we have for $|\partial^\alpha f(x)|$ when $|\alpha|=k$? For example, if ...
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1answer
30 views

Find $\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}$. What has gone wrong?

Find $\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}$. Using L'Hopital's rule I get that $$\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}=\lim_\limits{x\to 1}{\alpha x^{\alpha-1}\over \beta ...
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1answer
28 views

Conditions for a supremum of a set.

Suppose a function $f(x)$ is continuous on $[a, b]$ and there exists, $x_0 \in (a, b)$ such that $f(x_0) > 0$. And then define a set, $$A = \{ a \le x < x_0 \space | \space f(x) = 0 \}$$ We ...
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1answer
25 views

Show that ${\{s_n}\}$ is bounded if $s_n = b_1r + b_2r^2 + … + b_nr^n$ and $0 < r < 1$.

Question: Let ${\{b_n}\}$ be a bounded sequence of nonnegative numbers and r be any number such that $0 \leq r < 1$. Define $s_n = b_1r + b_2r^2 + ... + b_nr^n$ for every index $n$. Use The ...
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1answer
29 views

Prove $f(x+y)=f(x)+f(y)$ and $f(cx)=cf(x)$ [on hold]

if $f(x+y)=f(x)+f(y)$ with f continuous prove that $f(cx)=cf(x)$ for $c\in \mathbb{R}$ It has been asked before but my problem is that I have to prove it for Reals. But it seems impossible to me as ...
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1answer
15 views

Isometry on compact metric space?

Let $(X,d)$ be a compact metric space and also let $f:X\longrightarrow X$ be an isometry. Is it necessary that $f$ is onto? We know every Banach space can be regarded (or modeled) as $C(X)$ for ...
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2answers
99 views

To show sum of two sets is closed in R^2

Let $A=\{(x,y)\in \mathbb R^2:\max\{|y|,|x|\}\leq 1\}$ and $B=\{(0,y)\in \mathbb R^2:y\in \mathbb R\}$. Show that $A+B$ is a closed subset of $\mathbb R^2$ My try: let $z_n=x_n+y_n$ be a sequence in ...
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1answer
31 views

Showing continuity of a map

Let $(S, \rho)$ be a compact metric space. Suppose the map $T: S \to S$ is such that for every $x \neq y$, $$\rho(Tx, Ty) < \rho(x,y)$$ 1) Show that the map $\phi: S \to \mathbb{R}$ defined by ...
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24 views

Is a countable product of open intervals homeomorphic to $\mathbb{R}^\omega$?

Fix countably many intervals $(a_i,b_i) \subset \mathbb{R}$, and let $\pi_{i \in \mathbb{N}} (a_i,b_i)$ be their Cartesian product with the product topology. Question: is $\pi_{i \in \mathbb{N}} ...
2
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1answer
31 views

Proof Involving Generalized Mean

Let $x=(x_1,...,x_n) \in \mathbb R^n$ and $$g(p)=\sqrt[p]{\frac{1}{n}\sum_{k=1}^{n} |x_k|^p)}$$ Using Hölder's inequality, show that $g(p)$ is increasing on $(0,\infty)$. For a sequence with ...
3
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1answer
27 views

Proving a metric on X.

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= ...
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0answers
24 views

a valuable question about differential geometry, the curve$\beta(s)=\alpha(s)-rn(s)$

Let $\alpha(s)$,s$\in [0,l]$ be a closed convex plane curve positively oriented. The curve $$\beta(s)=\alpha(s)-rn(s)$$,where r is a positive constant and n is the normal vector, is called a parallel ...
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0answers
7 views

Generalized Holders inequality for infinity norm

The inequality is as follows: Let $g_j$ $j=1,..,n$ be measurable functions on a measure space $(X,M,\mu)$. Suppose $1 \leq p \leq \infty$ and $\sum_{j=1}^{n}\frac{1}{p_j} = \frac{1}{p}$ ...
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2answers
31 views

Every $p$-norm ($p \in [0,\infty]$) generates the same class of open sets on $\mathbb{R}^n$

The following claim has been made in my multivariable analysis class, and I think I have the idea of the proof but I can't quite seem to get down to the rigorous proof the instructor wants: Every ...
2
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2answers
39 views

Prove that there exists $x_0\in [a,b]$ such that $ \sum_{i=1}^{n} k_i \int_{x_0}^{x_i} fdt=0$

Let $f$ is a continuous function on $[a,b]$, $x_1,x_2,\ldots,x_n\in [a,b]$, $k_1,k_2,\ldots,k_n>0$. Prove that there exists $x_0\in [a,b]$ such that $$k_1\displaystyle \int_{x_0}^{x_1} ...
3
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2answers
41 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
2
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1answer
36 views

Integral inequality $\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$

Let $f(0) = 0$ and $0<f'(x)\leq1$ for all $x \geq0$, then prove: $$\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$$ The hint I was given was "differentiate, factor and ...
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0answers
26 views

Find $\lim_\limits{x\to \infty}{\sqrt[3]{x^3+ax^2+bx+c}-x}$. [duplicate]

Find $\lim_\limits{x\to \infty}{\sqrt[3]{x^3+ax^2+bx+c}-x}$. As I understand, I should use Taylor series, but I don't know how. What should I translate into Taylor series, to what extent, etc. This is ...
6
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1answer
60 views

Find $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$

Find $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$. I do know it is bounded by $1$. I tried using the sandwich rule with no ...
2
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1answer
50 views

Show that $f=0$ a.e. if $|\int_I f|^p \leq c|I|^{p-1}\int_I |f|^p$ with $0<c<1$

Suppose an extented real valued function $f$ defined on $\mathbb{R}^n$ satisfies the following two properties: a) There is a $p$, $1\leq p < \infty$ such that $f\in L^p(I)$, for every ...
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20 views

Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
3
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1answer
34 views

Sequence in an uncountable set of real numbers

Let $A$ be an uncountable subset of the real numbers, I think the following is true: There is an injective sequence $a:N\to A$ such that $\sum_{n=1}^\infty a_n$ diverges. This might also be true ...
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4answers
65 views

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$.

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you ...
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1answer
37 views

Proving Submultiplicativity on a Matrix Norm

Let $||A||=(\sum_{i=1}^{n}\sum_{j=1}^{n}{a_{ij}^p})^{1/p}$, and let p=2. Then prove that $\|AB\|\le \|A\|\|B\|$ I have looked at numerous proofs for this, and I don't see one that satisfies me ...
3
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1answer
24 views

If $f$ and $g$ are uniformly continuous on $\Bbb{R}$ then $f\circ g$ is uniformly continuous on $\Bbb{R}$

Prove or disprove: If $f$ and $g$ are uniformly continuous on $\Bbb{R}$ then $f\circ g$ is uniformly continuous on $\Bbb{R}$. I think there's something crooked in my attempt. I would like to know what ...
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2answers
42 views

Expression for Taylor's formula with a remainder

Assume $f$ has a continuous second derivative $f~''$ in some neighborhood of $a$.Then, for every $x$ in this neighborhood, we have $f(x) = f(a) + f~'(a)(x-a) + E_1(x)$ , where $E_1(x) = \int_a^x ...
1
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1answer
35 views

Extending Minkowsky inequality to double summation?

I know the Minkowski inequality for sequences as follows : $$\left(\sum_{k=1}^n|x_k+y_k|^p\right)^{1/p} \leq \left(\sum_{k=1}^n|x_k|^p\right)^{1/p}+\left(\sum_{k=1}^n|y_k|^p\right)^{1/p}$$ Now say we ...
1
vote
1answer
27 views

Prove an altered p-norm is increasing

$x=(x_1, x_2, \ldots, x_n)$ Prove that $g(p)=[(1/n)(\sum_{k=1}^n |x_k|^p)]^{1/p}$ is increasing on the interval $(0, \infty)$, and find $\lim_{p\to\infty}g(p)$ I find this is extremely difficult. I ...
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0answers
8 views

Riemann Integration - Upper Sum

Let $P = \{x_o, x_1, ..., x_n\}$ and $Q = \{x_o, x_1, ...x_j, z, x_{j+1}, x_n\}$ be partitions of $[A, B]$. Note that Q is a refinement of P with just one extra point. Show that if $f: [A,B] \to ...
2
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0answers
41 views

Fourier transform of $e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} $

I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of ...
4
votes
2answers
113 views

Real solutions of $x^n + y^n = (x+y)^n$

I have to find all real solutions of the following equation: $x^n + y^n = (x+y)^n$ Clearly for $n = 1$, the equation holds for every $x,y$ real numbers. If $n$ is greater or equal to $2$, we do ...
2
votes
5answers
80 views

Limit problems and quandaries: finding $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$.

Find $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$. What I did is: $\lim_\limits{n\to \infty }{({n^2-n\over n^2+1})^{n+10}}=\lim_\limits{n\to \infty } {({n^2+1-1-n\over ...
1
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1answer
32 views

Prove that $\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$

If $f_n(x)=nxe^{-nx^2}~\forall~n=1,2,\cdots$ and $x$ real, show that $$\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$$ Attempt: By the $Mn$ Test, it ...
1
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2answers
29 views

Does this supremum equal infinity?

This is a generalization of the previous question Does this infinum tend to infinity? Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function satisfying $$\sup ...