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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55 views

An Approximation involving the Exponential Integral

Define for real $x > 0$ the function: \begin{equation} F(x)= 1 + x e^{x} Ei(-x), \end{equation} where $Ei(x)$ is the exponential integral. I found in a physics papers (Amaldi, Fluctuations in ...
2
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67 views

Question about an infinite product

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
2
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1answer
72 views
+50

Drawing large rectangle under concave curve

Let $f$ be a continuous concave function on $[0,1]$ with $f(1)=0$ and $f(0)=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k\cdot \int_0^1f(x)dx$, ...
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27 views

General solution to an trigonometric equation set

An extent for last question I asked is to solve $$\begin{cases} 2\cos(2y_1)=1+\cos(y_1+y_2)\\ 2\cos(2y_2)=\cos(y_1+y_2)+\cos(y_2+y_3)\\ 2\cos(2y_3)=\cos(y_2+y_3)+\cos(y_3+y_4)\\ ...\\ 2\cos(2y_{n-1})=\...
6
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4answers
1k views

Intersection of infinite sets is infinite?

I know that if $C \subseteq [0,1]$ is uncountable, then there exists $a \in (0,1)$ such that $C \cap [a,1]$ is uncountable. Is it still true for any infinite sets? That is, if $C \subseteq [0,1]$ is ...
2
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2answers
85 views

Does there exist a surjective function from $(0,1)$ to $[0,1]$?

Can I say that cardinality of $(0,1)$ is less than the cardinality $[0,1]$ ?
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66 views

Drawing large rectangle under curve

Let $f$ be a continuous nonincreasing function on $[0,1]$ with $f(1)=0$ and $\int_0^1 f(x)dx=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k$, with ...
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17 views

Differentiable sub manifolds and regular parametrization

Let $0<r<R$. Consider the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \mid (\sqrt{x^2+y^2}-R)^2+z^2=r^2\}.$$ How can I show that $T^2$ is a two-dimensional differentiable submanifold of $\mathbb ...
2
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1answer
57 views

Conclusion about Convergence of Series

Assume that $$ \sum_{n=1}^{\infty}a_n < \infty$$ and $a_n\geq0$. What can be said about the asymptotic behavior of $a_n$ then? For example, I would have liked to say $$a_n<\frac{1}{n}$$for all $...
8
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2answers
246 views

Is there a nonabelian topological group operation on the reals?

Inspired by A binary operation, closed over the reals, that is associative, but not commutative. That question asks for a noncommutative semigroup operation on $\Bbb R$, for which right projection is ...
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1answer
58 views

Nonempty closed sets on a connected space imply nonemptiness of intersection?

I am dealing with just real line to make things little easier for me. Suppose we have a set $X=[0,x],X'=[x,\infty)$. For the sake of argument, assume both are closed and nonempty. Claim: By the ...
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2answers
41 views

Real analysis: Continuity and Differentiability [on hold]

Let $f(x)=x^2$ if $x$ is rational and $f(x)=0$ if $x$ is irrational. a) Prove that f is continuous at exactly one point, namely $x=0$. b) Prove that f is differentiable at exactly one point, namely $...
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1answer
41 views

Prove that addition preserves order. (for natural numbers)

Prove that addition preserves order. $a ≥ b$ if and only if $a+c ≥ b+c$. (using peano axioms) I try to do it by induction on $c$. Can I use $(a+c)++ ≥ (b+c)++$. I am not sure because first we will ...
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3answers
132 views

How to integrate $\int _1^{\infty }\frac{dx}{\left(x^2+1\right)\sqrt{x^2-1}}= \;?$

How do I integrate $\int _1^{\infty }\left(\frac{1}{\left(x^2+1\right)\sqrt{x^2-1}}\right)\:dx$? So what I've tried is substituting $x\:=\:\frac{1}{\sin t}$. So then I'll have that when $x\rightarrow ...
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1answer
21 views

Problem in conceiving the derivation of 'Limit of Sequence to Zero Distance Point'.

I was following the derivation of 'Limit of Sequence to Zero Distance Point'; here is the excerpt of the derivation: First it is shown that: $$\forall n \in \mathbb N_{>0}: \exists x_n \in S: \...
3
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1answer
95 views

Show that $f(x)=\frac{2}{x}\cos \frac{1}{x^2}$ is NOT bounded on [-1,1]

Show that $f(x)=\frac{2}{x}\cos \frac{1}{x^2}$ is NOT bounded on [-1,1] The hint given in the book: Let $a_{n} = \sqrt{\frac{2}{(2n-1)\pi }} $ Then $\cos (1/a_{n}^2)=1$ for all n, $a_{n}\rightarrow ...
3
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3answers
140 views

Continuous function satisfying $f(x+1) = f(x)$

Let $f : \mathbb{R} \to \mathbb{R} $ be a continuous function and $f(x+1) = f(x)$ for all $x \in \mathbb{R}$. Then $f$ is bounded above, but not bounded below. $f$ is bounded above and ...
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1answer
70 views

What's the order of operations when dealing with function composition?

Given $f:[0,1]\rightarrow \mathbb{R}$ and $g:[0,1]\rightarrow [0,1]$, $g(x)=x^2$. Which of the two equalities is true? 1)$f^2(x^2)=f^2(g(x))=(f^2\circ g)(x)$; 2)$f^2(x^2)=f(x^2)\cdot f(x^2)=f(g(x))...
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1answer
81 views

Continuous injection and density in $l_p$ spaces

If $r \le s$ then $l_r$$\subseteq$ $l_s$ . How can I prove there is a continuous injection $l_r$ $\hookrightarrow$ $l_s$? The suggestion was to use the fact that $\Vert$x$\Vert$$_r$ $\le$ $\Vert$x$\...
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19 views

differentiation under the integral sign and change of variables

Let $f \in C^2 (\mathbb{R}^2)$ with a bounded support, and let $f_\phi (x,y)=f(x\cos{\phi}-y\sin{\phi},x\cos{\phi}+y\sin{\phi}))$ show that: $\frac{d}{d\phi}\iint_{\mathbb{R}\times(0,\infty)}f_\phi(...
3
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1answer
119 views

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable?

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable? To clarify: the problem stated that the composition is well ...
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23 views

Proof explanation: diam(cl(E)) = diam ( E)

Yesterday I posted a question on this site about proving diam($E$) = diam(cl($E$)). A user posted his proof but there was some part that I don't get, I commented on his proof but he didn't respond ...
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0answers
38 views

How do I prove that $L^1$ functions are almost continuous?

Let $f\in L^1(\mathbb{R}^n)$ and let $\epsilon >0$. How do I prove that there exists a measurable $A\subset \mathbb{R}^n$ such that $m(\mathbb{R}^n \setminus A)<\epsilon$ and $f\upharpoonright (\...
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2answers
68 views

One example: $f'(a)$ isn't invertible function, then $(f^{-1})'(f(a))$ isn't invertible.

I would like to find one example: let be $f:A\subset \mathbb{R}^p \to \mathbb{R}^p$ and $g:f(A) \to \mathbb{R}^p$ its inverse function. Supose that $f$ is differentiable in $a\in A$ and $g$ in $b= f(...
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1answer
60 views

Radon Nikodym Thm: extending to $\sigma$-finite case

I am reading Bartle's "Elements of Integration". Radon-Nikodym Thm: Let $\lambda,\mu$ be $\sigma$-finite measures on a measurable space $(X,\textbf{X})$ and say $\lambda \ll \mu$. Then $\exists$ ...
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3answers
87 views

Does the series $S(x) = \sum_{n=1}^\infty \frac{n^2x^2}{n^4+x^4}$ converge uniformly?

does the above series uniformly converges? If it does, how to find the interval of x in which it uniformly converges? $ {(n^2-x^2)}^2 \ge 0 \Rightarrow n^4 + x^4 - 2n^2x^2 \ge 0$ This gave me $\frac{...
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2answers
88 views

If $\int_0^{2\pi} q = 0$, then $\lim_{n \to \infty} \int_0^{2\pi}p(x)q(nx) = 0$

I'm learning about Fourier series and need help with the following exercise: Let the functions $p, q \in L^1([0, 2\pi])$ be bounded and $2\pi$-periodic. If $\int_0^{2\pi} q = 0$, show that $...
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1answer
36 views

How to derive the existence of (a+b+c) in a set containing those elements

I am looking to derive the existence of quantity (a+b+c) given that so far I only have the group axioms and knowledge of the sum of two quantities from the Associative Law: a + (b + c) = (a + b) + c. ...
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1answer
41 views

Comparison of capacity of sets in $\mathbb{R}^n$

This is mainly in reference to this MSE post. Let $B_r \subset \mathbb{R}^n$ denote the ball of radius $r$ centered at the origin. Consider any set $F \subset B_1$. For all sets $\Omega \subset \...
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2answers
61 views

Definition of an Ordered Field

A text I'm looking at has the following definition of an ordered field: DEFINITION A field ($F$, $+$, $\cdot$) is ordered iff there is a relation $\lt$ on $F$ such that for all $\quad\quad\quad\...
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48 views

Constructing a null set and a Lipschitz function nowhere differentiable on it

I'm trying to solve the following exercise. Now, Rademacher's theorem says that locally Lipschitz functions are $\mathcal L^N$-a.e. differentiable, so $E$ must be a null set, and this is clearly ...
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1answer
21 views

finding a close enough point with implicit function theorem

Let $a_1, ...,a_n ,B\in \mathbb{R}^n$ , not all on the same plane. Prove that for a small enough neighborhood of zero $U$ and $\forall u_1,...,u_n \in U $ there is a point $C \in \mathbb{R}^n$ that $...
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4answers
126 views

Can $\int_0^{2\pi} \frac{dx}{\sin^6x+\cos^6x}$ be solved using $\cot x = u$ as substitution?

My first guess is it can't, since when I substitute the boundaries, I end up with $\cot2\pi$ and $\cot0$. Nevertheless I tried substituting pretending it is indefinite integral, but I can't get ...
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0answers
46 views

An infinite product for $\arccos \alpha$ similar to Viete's formula for $\pi$

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
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0answers
32 views

Function sequence and some properties

Consider functions $f_n$, $f : \mathbb{R} \rightarrow \mathbb{R}$ such that the sequence $\{f_n\}$ is uniformly convergent to $f$ and every $f_n$ has property $W$. Determine whether $f$ must have $W$ ...
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2answers
45 views

problem in real analysis about open sets in metric spaces.

For $x = (x_1, x_2, \ldots, x_n)$ and $y = (y_1, y_2, \ldots, y_n)$ in $\mathbb{R}^n$. Let $d_p(x, y) = \Bigg(\sum\limits_{i=1}^n |x_i-y_i|^p\Bigg)^\frac{1}{p}$ for $1 \leq p < \infty$ and $d_\...
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2answers
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for $f\in C^2(\mathbb{R})$, finding the derivative of $\frac{d}{dt}\int_0^\infty f(x+t)\cdot xdx$

Let $f\in C^2(\mathbb{R})$, (a) Prove that $$\frac{\mathrm{d}}{\mathrm{d}t}\int_0^\infty f(x+t)\cdot x\mathbb{d}x=-\int_0^\infty f(x)\mathrm{d}x$$ (b) Prove that $$ \iint_{(0,\infty)\times(...
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1answer
36 views

What is the relation between the matrix of an operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, either both real or both complex, and let $T \colon X \to Y$ be a linear operator. (Then $T$ is bounded since its domain is finite-dimensional). ...
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52 views

How much can the integrability at zero tell about the decay rate around zero?

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
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1answer
26 views

Prove $\varphi \colon (1-\varepsilon,1+\varepsilon) \to \mathbb{R}$ exists.

Prove that for sufficient small $\varepsilon$ there exists a two times differentiable function $\varphi \colon (1-\varepsilon,1+\varepsilon) \to \mathbb{R}$ such that $\varphi(1)=0$ and $$ \cos (\...
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1answer
73 views

$\int_{- \infty}^{+ \infty} |f(t)| dt < \infty \implies \int_{-\infty}^{x} f(t) dt$ is continuous?

I've found counter example for $(A),(D)$ and have shown except a bounded interval $F$ is uniformly continuous everywhere else. And so $(B)$ would imply $(C)$ is correct. But I can't show $(B)$ is ...
5
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3answers
138 views

Isometry map on a compact metric space

Let $X$ be a compact metric space and $f : X\rightarrow X$ such that $d (x,y)\le d (f(x),f(y))$ for all $x,y\in X$. Prove that $f$ is an isometry. I am getting stuck on this question. Can any one help ...
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19 views

Prob. 5, Sec. 4.5 in Kreyszig's functional book: The adjoint of the composite of two bounded linear operators

Let $X$, $Y$, and $Z$ be normed spaces, either all real or all complex. Let $T \colon X \to Y$ and $S \colon Y \to Z$ be bounded linear operators. Let $X^\prime$, $Y^\prime$, and $Z^\prime$ denote the ...
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1answer
34 views

Partition of unity from RCA Rudin

Let me ask the following question: How Rudin applies Theorem 2.7 in the begining? He take some $x\in K$ then $x\in V_i$ where $i=i(x)$. What's next? I thought on this about couple hours but no ...
0
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1answer
57 views

Extrema of $f(x)=tx^2+e^{-x}$

I cannot seem to find the extrema of the function $f(x)=tx^2+e^{-x}, t>0$. The function's graph suggests that there exists a minimum. The derivative is $f'(x)=2tx-e^{-x}$ and $f'(x)=0 \...
0
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0answers
44 views

Integral evaluation - Gamma distribution

I have a sequence of independent random variables which are $\chi^2(1)$ distributed, $(X_i)_{i=1}^n$, $X_i\sim\chi^2(1)$. If I consider the sum $\frac{t}{n}\sum_{i=1}^n{X_i}$ this should be $\sim\text{...
2
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1answer
36 views

Classification of an open set in real

Prove that open set in real line can be represented as ar most countable disjoint union of open intervals. I know that this question repeated many times in MSE but let me ask the following question. ...
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0answers
60 views

Urysohn's Lemma from RCA Rudin

I found out the proof of Urysohn's Lemma from Rudin's book but I have couple questions which I am not able to answer. 1) Why Rudin wrote that "in terms of characteristic functions, the conclusion ...
1
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2answers
38 views

Prove that order is antisymmetric. (for natural numbers)

Prove that order is antisymmetric.(for natural numbers)i.e. If $ a \leq b$ and $b\leq a$ then $a=b$. I do not want a proof based on set theory. I am following the book Analysis 1 by Tao. It should be ...
3
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2answers
80 views

Can $\{(f(t),g(t)) \mid t\in [a,b]\}$ cover the entire square $[0,1] \times [0,1]$ ?

Suppose $f,g : [a, b] \to R$ are both continuous and of bounded variation. Then can set $\{(f(t),g(t)) \mid t\in [a,b]\}$ cover the entire square $[0,1] \times [0,1]$ ? And what if we remove the ...