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, integration through the fundamental theorem of calculus.

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Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
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1answer
39 views

Divergence-Convergence of the sequence $\sin(n!{\pi}\theta)$

I am working on the convergence-Divergence of $\sin(n!{\pi}\theta).$ In his book, Hardy(A Course of Pure Mathematics) page 128 cited " The case in which $\theta$ is irrational cannot be dealt with ...
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1answer
33 views

Proving the Weierstrass M-Test with topology

I've encountered some theorems in analysis that are ultimately provable in a more elegant way with topology. So, is there a topological proof of the Weierstrass M-Test, ideally not using terribly ...
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46 views

Continuity of a piecewise constant function

A)I can draw the graph and see that the function is continuous at x=0.3 as when you approach it from the left and right you get the same result B) not sure how to prove properly but it is not ...
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1answer
17 views

Is this theorem about “completion of metric space” correct?

It's well-known that there is a completion of a metric space unique upto isometry. I have tried to modify this theorem slightly and I proved this statement: Let $(X,d_X)$ be a metric space. ...
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23 views

Convergence of this Sequence of Functions to its Supremum [closed]

Let f be positive and continuous on $J=[a,b]$. Let M=sup{ f(x) : x $\in$ J}. Show that : $$ M=\lim_{n}\left ( \int_{a}^{b}(f(x))^ndx\right )^{\frac{1}{n}}$$
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31 views

Does the Fourier coefficients of a function $f\in H^1(0,L)$ (the first order Sobolev space) are absolutely summable?

My precise question: Let $f\in H^1(0,L)$ and let $\{f_n\}$ be its Fourier sine series coefficients on $(0,L)$, is it true that $\{f_n\}\in l^1$, i.e. $$\sum_{n}|f_n|< \infty .$$ Thanks
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constrained optimization and differential equation

Consider the following differential equation system (cylindrical coordinate system): $\frac{dP_x}{dz} = P_x C \int\limits_0^{2\pi}\int\limits_0^a \frac{f(r, \theta)}{g(r, \theta, z)} r dr d\theta$ ...
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1answer
48 views

Is liminf of a product equal to the product of liminfs?

My question is just for curiosity. I was thinking if is true this curious affirmation: Let $a_n$ a bounded sequence of nonnegative numbers and $b_n$ a convergent sequence of negative numbers. Then ...
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2answers
42 views

If a continuous function is positive at a point, it is also positive in some neighborhood of the point [closed]

Suppose that $f:\mathbb{R}^k\to\mathbb{R}^1$ is a continuous function and that $f(x^*)>0$. Show that there is a ball $B=B_\delta(x^*)$ such that $f(x)>0$ for all $x\in B$.
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27 views

$l^1$ and $l^{\infty}$ not isometric in general [duplicate]

I'm trying to figure out the following example: Consider the map $\Lambda:l^1\rightarrow (l^{\infty})^{\ast}$ given by $$\Lambda(\xi)(\eta_1,\eta_2,\cdots)=\sum_i \xi_i\eta_i$$ I can show that it is ...
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1answer
31 views

If $f \in L^{1}(d\mu)$, is it true that $\int \limits_{X} f\chi_{\{ f \neq 0 \} } \,d\mu = \int \limits_{X}f \,d(\chi_{\{ f\neq 0 \} }\,d\mu)$?

Ok, so we have $f \in L^{1}(d\mu)$, with $(X, \Sigma, \mu)$ a complete measure space. If we assume $f$ is nonnegative, we can define a measure $\rho(E) = \int \limits_{E} f \,d\mu$ for $E \in ...
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2answers
69 views

Prove that if an infinite series converges, then the associative property holds

I'm self-studying from the book Understanding Analysis by Stephen Abbott and have no idea how to do exercise 2.5.2 on page 57. The exercise is as follows: Prove that if an infinite series converges, ...
3
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1answer
40 views

If $f \in L^{1}(d\mu)$ is nonnegative, can we conclude $\mu( \{ x \mid f(x) \neq 0 \} ) < \infty$?

I am trying to prove a statement, and I need the fact that: If $f \in L^{1}(d\mu)$ is a nonnegative function, then this implies $\mu( \{x \mid f(x) \neq 0 \} ) < \infty$. But I don't know ...
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1answer
62 views

How to express $x^5$ as a telescoping series

How do I express $x^5$ as a telescoping series (i.e, $x^5=x^5-x^{a}+x^{a}-x^b+x^b-...)$? In other words, I must find functions $b(n),c(n),d(n)e(n)$ such that ...
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1answer
33 views

Monotone increasing function has finite one-sided limits at every point

"Let $f:(a,b)\rightarrow\mathbb{R}$ be a monotonous increasing function on $(a,b)$ and let $x_0\in(a,b)$. Show that $f$ has finite limits $f(x_0^-)=\lim_{x\rightarrow x_0^-}f(x)$, ...
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2answers
42 views

Integrable function with given condition is in $L^p$

Suppose $f:\Bbb R \to \Bbb R$ is integrable and there exist constant $c\gt 0$ and $\alpha \in (0,1)$ such $$\int_A |f(x)|dx\le cm(A)^\alpha$$ for every Borel measurable set $A\subset \Bbb R,$ where ...
3
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1answer
47 views

Measure Theory Inequality

I was having trouble showing the following inequality: Prove that if $A \subset I = [0,1]$ has measure $u(A) < 1$ and $\epsilon > 0$, then there is an interval $[a,b] \subset I$ such that $u(A ...
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1answer
29 views

Lebesgue measurable functions and the absolute value of them

Let $f$ is a measurable function. If $f$ is Lebesgue integrable, is the absolute value of $f$ Lebesgue integrable?
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118 views

Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

It's easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational. However, can it be shown whether it is algebraic or transcendental? My hunch is that it's transcendental but I don't know ...
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1answer
26 views

A “repeated roots allowed” version of the continuity of roots

Let $R_n$ denote the set of all monic real polynomials of degree $n$ all of whose roots are real. Then $R_n$ is a closed subset of the $n+1$-dimensional space ${\mathbb R}_n[X]$. For $P\in R_n$, ...
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15 views

Fourier series and irrational period

I'd appreciate it if I could get a hint to solve the following exercise. Let $\alpha$ be an irrational number and let $f$ be a real valued measurable function on $\mathbb{R}$ such that ...
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1answer
65 views

Analytical solution of a polynomial $a\cdot x^{e}+b\cdot x^{4\cdot e}+c =0$

Is it possible to get an analytical solution of the equation $a\cdot x^{2\cdot e}+b\cdot x^{e+1}+c =0$ Which can be also written as (due to the value of $e$): $a\cdot x^{e}+b\cdot x^{4\cdot e}+c ...
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2answers
48 views

$|f(x)-f(y)|<|x-y|$ on a non-empty closed bounded set of real numbers

Let $A$ be a non-empty closed bounded set of real numbers and $f: A \to A $ be a function such that $|f(x)-f(y)|<|x-y| , \forall x,y\in A$ , then how to show that $f$ has a unique fixed point ?
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1answer
38 views

Does every convergent sequence have a sub-sequence whose terms comes closer than any positive sequence?

Let $(x_n)$ be convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers , then is it true that there is a sub-sequence $(x_{r_n})$ such that ...
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0answers
28 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
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29 views

Alternative derivation of Poincaré inequality

I've been trying to prove the Poincaré inequality via a representation formula for Sobolev functions $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p < n$, wlog with compact support. The setting is ...
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1answer
35 views

Is Banach space a correct context to study sequences and series?

Recently, I have reviewed elementary analysis and I realized that every theorem in the text(Rudin-PMA) about series can be generalized to Banach space. Here is an example. Below is the theorem ...
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3answers
353 views

Determining if a recursively defined sequence converges and finding its limit

Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists. I ...
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1answer
29 views

Criteria for measure convergence implying convergence a.e.

Suppose the function $g_n = \sup_{m \geq n} |f_n-f_m|\to 0$ in measure. Show $f_n \to f$ a.e. Suppose instead that $\sum_{n=1}^{\infty} m\{ |f_n - f|>\epsilon\} < \infty$. Show $f_n \to f$ a.e. ...
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76 views

For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof

For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof I'm following a book and it just uses this, it doesn't say anything about the function, so I've not assumed it's ...
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1answer
60 views

Point-wise converging convex functions on $[0,1]$

Suppose we have a sequence of continuous convex functions $\{f_n\}$ defined on $[0,1]$ which converge point-wise to a limit $f$ on $[0,1]$, i.e. for all $x \in [0,1]$ $$\lim_n f_n(x) = f(x).$$ Let $G ...
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99 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
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1answer
29 views

If $E, F \subset [0, 1]$, $m(E), m(F) > 0$, and $E_n = \{x \in [0, 1] : nx \bmod 1 \in E\}$, show $m(F\cap E_n) > 0$ for sufficiently large $n$

Suppose $E \subset [0, 1]$ has positive Lebesgue measure and let $E_n = \{x \in [0, 1] : nx \bmod 1 \in E\}$. If $F \subset [0, 1]$ has positive Lebesgue measure, show that so does $F \cap E_n$ for ...
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1answer
39 views

Inequality involving functions bounded above but not below: $|\sup_{x\in\mathbb R}f(x)-\sup_{x\in\mathbb R}g(x)|\leq\sup_{x\in\mathbb R}|f(x)-g(x)|$?

Let $ f,g: \mathbb{R} \rightarrow \mathbb{R} $ be bounded above and below, then we can prove that: $$ | \sup_{x \in \mathbb{R} } f(x) - \sup_{x \in \mathbb{R} } g(x) | \leq \sup_{x\in \mathbb{R}} | ...
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1answer
55 views

Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates

Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates $$ \left\{ \begin{array}{c} u = yz\sin(x)\\ v = y^2 - x\\ w = xz \end{array} \right. $$ Determine the ranks ...
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3answers
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$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping.

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping $f$ cannot be one-to-one mapping. Let $D_1F(x,y) \neq 0$ for all $(x,y)$ for some open ...
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Calderón reproducing formula : $\int_{0}^{\infty}\int_{R^d}|\phi_{t}(x-y)||(\phi_t*f)(y)|\frac{dt}{t}dy<\infty$

Suppose that $\int f=0$, $f \in L^2$ and $f$ has a compact support. Let $\phi$ be radial, and such that $\mathrm{supp}(\phi) \in B(0,1)$. Plus, assume that $\int_{R^+} ...
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30 views

math formulation of splitting intervals algorithm

So I have this recursion in my mind I am trying to formulate in a sequence fashion for some algorithm problem. So given i , I can tell you where $x_{i}$ along the real line (like the type writer ...
2
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1answer
33 views

Function sequence $f_n(x)=n^{\alpha}x^3e^{-nx^2}$

Let $f_n(x)=n^{\alpha}x^3e^{-nx^2}$ for $x\in[0,1]$ I'm being asked for which $\alpha$ is $f_n$ pointwise convergent and for which is it uniformly. If I'm correct ...
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0answers
36 views

A question on two sequences of real functions

Define $$S\left( \alpha ,C \right)=\left\{ f:\mathbb{R}\to {{\mathbb{R}}^{+}}:\int_{\mathbb{R}}{f\left( u \right)du}=1,\,\,\int_{\mathbb{R}}{{{\left| {{f}^{\operatorname{ft}}}\left( t \right) ...
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60 views

Find the minimum value of $t^3+t^2-2t-2$ given that $t$ is greater than or equal to 2

The original question was to find the range of the function f defined by: $$f(x)=\frac {(1+x+x^2)(1+x^4)}{x^3}$$ for $x>0$ Evidently, differentiating is not very helpful. So I wrote $f(x)$ as: ...
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2answers
49 views

An ABC soft question about epsilon-delta argument

Someone told me that some textbooks present epsilon-delta argument somewhat misleadingly. For example, consider the simplest one: the convergence of the sequence $(1/n)_{1}^{\infty}$ to $0$. These ...
4
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1answer
42 views

What exactly are the curves that are a best fit to the Harmonic Cantilever?

Let's start with a few references to get an idea: Daniel Goldwater: Harmonic Cantilever Book Stacking Problem Block-stacking problem Harmonic Series and Bricks Interesting related issues: Maximum ...
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3answers
40 views

A uniform continuity problem

Let $A$ be a set of real numbers and $f:A \to \mathbb R$ be a function such that for every $\epsilon >0$ , there exist a uniformly continuous function $g_\epsilon :A \to \mathbb R$ such that ...
2
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3answers
196 views

Why are variables in integration by substitution so counter intuitive?

Integration by substitution is defined as something like $\displaystyle\int_a^b f(\phi(t))\phi'(t)dt = \int_{\phi(a)}^{\phi(b)} f(x)dx$ But for my taste, the variables $x$ and $t$ are exactly ...
2
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0answers
49 views

$f\in C^\omega ((a-R,a+R),\mathbb{R})$ [closed]

We discuss the following question in the field of real numbers . A a power series $f(x)= \sum_{n=0}^\infty a_n (x-a )^n$ converges in $(a-R,a+R).$ Prove: $$\forall x_0\in\left(a-R,a+R \right), ...
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0answers
30 views

Prove that every subset of $\mathbb{R}$ is compact in the finite complement topology.

I need help with my proof in particular. I am aware that there is a similar question elsewhere. Can someone please verify my proof or offer suggestions for improvement? Prove that every subset of ...
3
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4answers
104 views

Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$?

It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as: $$ \begin{align} f : (a,b) &\to (\alpha,\beta) \\ x &\mapsto \alpha+(x-a)(\beta-\alpha). ...
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1answer
32 views

Bolzano–Weierstrass theorem for random variables?

I am wondering if there is something similar to the Bolzano–Weierstrass theorem for random sequences. Namely, let $\{x_n\}$ be a bounded random sequence. Is it true that, under some reasonable ...