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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10 views

Rational Numbers. How to prove?

$Given :\ u+v \ is \ rational, \ u^2 + v^2 =1 \ , prove \ v^n + u^n \ is \ rational$. What i have done so far is proving that $uv$ is rational by expanding $(u+v)^2$. I expanded $(u+v)^n$ using ...
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22 views

Inequality $\left(\sum\limits_{k=1}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=1}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$

I am trying to find a proof of the following inequality $\left(\sum\limits_{k=1}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=1}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$. Does ...
3
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2answers
23 views

Showing that $(f_1,f_2,\dots,f_m)$ is a measurable function from $(\mathbb R^m,\mathcal B(\mathbb R^m))$ into itself

If $\{f_i, 1 \le i \le m\}$ is a set of real valued Borel functions on $\mathbb R$, how to show a vector of functions, $(f_1, f_2,..., f_m)$ is a measurable mapping from $(\mathbb R^m, ...
2
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1answer
34 views

Boundary of a bounded open set in $\mathbb{R}^2$

Does the boundary of a bounded open set in $\mathbb{R}^2$ necessarily have infinite points? How do we prove that, or is there a counterexample? It seems true to me, but I haven't been able to find a ...
2
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0answers
17 views

If $\lambda=$ measure of a set and all $G_k$'s are open sets, then : $\lambda ( \cup_{k=1}^{\infty} G_k ) \le \sum _{k=1}^{\infty}\lambda ( G_k)$

I just started reading the book Lebesgue Integration on Euclidean Spaces by Frank jones, in which the author gives a result and it's proof as : the If $\lambda$ denotes the measure of a set and all ...
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1answer
27 views

if $a,b,c,d \in \mathbb R$ such that $a < b$ and $c < d$, then prove that $[a,b]$ is equivalent to $[c,d]$.

What am I supposed to do? I'm relearning cardinality of sets, Archimedean property, infimum and supremum...
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0answers
13 views

Max function as bounded functions

I have an algebra of bounded functions $A$ that contains the constant functions and is closed under uniform convergence. We also have that if $f \in A$ then $|f| \in A$. I'm trying to show that if $f, ...
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1answer
50 views

Folland, Real Analysis Theorem 1.19

Theorem: If $E\subset\mathbb{R}$, the following are equivalent a.) $E\in M_\mu$ b.) $E = V\setminus N_1$ where $V$ is a $G_\delta$ set and $\mu(N_1) = 0$ c.) $E = H\cup N_2$ where $H$ is a ...
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1answer
17 views

Approximation of continuous functions by Bernstein polynomials

Recently a professor show me the following heuristic to provide approximations of continuous functions by polynomials: Let $P_n(x) = \sum_{k=0}^{n} {n \choose k} f(\frac{k}{n}) x^k (1-x)^{n-k}$. ...
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1answer
28 views

Finding the adherent points of $A=\left\{\left(1/n,1/m\right)|n,m\in\mathbb{N}\right\}$

The obvious adherent point is $(0,0),$ then I thought about fixing a point for each component and finding the adherent points on each line, but it leaves a mess. Doing it "my way" would lead to find ...
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1answer
23 views

Trying to construct a specific function

I am trying to construct a function $f$ with the following property: $\mathbf{N}$ is the set of natural numbers without 0. Show that $\forall \epsilon>0: \forall a,b \in \mathbf{N}: a < b: ...
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2answers
43 views

Convergence of a sequence of convolutions

Let $(a_n)$ be a sequence of real numbers such that $$ a_0>a_1>\cdots>0 $$ and $M:=\sum_{n=0}^\infty a_n<+\infty$. Denote $$ g_n=\frac{1}{a_n}\cdot 1_{[0,a_n]} $$ and define ...
2
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3answers
82 views

$f:\mathbb R^{2} \rightarrow \mathbb R$ s.t ${f(x,y)}={{xy}\over {x^{2}+y}}$ is not continuous at the origin

$f:\mathbb R^{2} \rightarrow \mathbb R$ is defined as $${f(x,y)}={{xy}\over {x^{2}+y}}$$; when $x^{2}+y\neq 0$ and $$f(x,y)=0$$ otherwise. To show this is not continuous at the origin . ...
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1answer
29 views

When is it ok to use a sequential limit in place of a continuous limit?

I am working through some Lebesgue integral problems, and I've come across a few instances where I would like to use the dominated/monotone convergence theorems, but the limit is continuous, and I'm ...
3
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1answer
20 views

All closed rational rays measurable implies $f$ measurable

Is the following proof correct? Let $f: X \to \mathbb{R}$ where $X$ is a measurable space. Suppose $\{x: f(x) \geq r\}$ is measurable for each $r \in \mathbb{Q}$. Then, $f$ is measurable. Proof: ...
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1answer
29 views

Divergent or convergent but how ??

I was to depict the convergence & divergence nature of the summation $\sum A_n$ where $A_n = (n^{1/n}-1)^k$ I was able to prove that when $k>1$ then $\sum A_n$ is converging and while $k<0$ ...
5
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1answer
35 views

Show that the image of Lipschitz function $\gamma : [0,1] \to R^n$ has measure $0$, if $n \ge 2$.

Problem Statement: Let $\Gamma$ be the image of a Lipschitz continuous function $\gamma : [0,1] \to R^n$, that is, $\Gamma = \{\gamma(t) : t \in [0,1]\}$, and $|\gamma(t_1) - \gamma(t_2)| \le K |t_1 - ...
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0answers
19 views

Bound on the mean value of function involving Hilbert transform

Consider the integral $$\int_{-\infty}^{\infty} x|A|^2_x\mathbb{H}(|A|^2_x) \ dx,$$ where $A=A(x,t)$ is a complex valued, compact function (I mean this in the heuristic sense that $A$ vanishes ...
3
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4answers
51 views

Proving convergence of a series and then finding limit [duplicate]

I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$ converges and find the limit. I started by defining the sequence by $x_1=\sqrt{2}$ and then ...
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31 views

Difficulty in understanding converse part of proof of a propostion in Andrew Browder's Mathematical Analysis

Proposition: Let $\mu$ be finitely additive set function, defined on the algebra $\mathscr A$. Then $\mu$ is countably additive if and only if its has following property: if $A_n \in \mathscr A$ ...
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2answers
36 views

Question on Taylor series in real analysis

Suppose that $$ f(x) = \begin{cases} e^{-1/x^2} & \text{if }x\ne 0, \\ 0 & \text{if }x=0. \end{cases} $$ How do I prove that $(d/dx)f$ at $0$?? I tried it this way, \begin{align} f'(0) & = ...
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16 views

When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

When are $\frac{1}{|x|^s}$ for $s>0$ and $\log|x|$ integrable near the origin? I'm reading Evans PDE and in the construction of the fundamental solution of Poisson's equation, he defines $$ \Phi(x) ...
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0answers
19 views

Is the function ${e^{-{1}\over{x}}}\over {x}$ on $(0,1)$ uniformly continuous or bounded?

$$f(x)= {{e^{-{1}\over{x}}}\over {x}}$$ for $x\in (0,1)$ . Is this function $a$) uniformly continuous $b$) bounded but not continuous $c$) unbounded This would be uniformly ...
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1answer
28 views

Solution to the wave equation in $\mathbb{R}^{3}$ with certain initial data

Suppose $f$ is a smooth function satisfying $f(0) = f'(0) = 0$. The question I am working on is to determine the solution $u$ to $u_{tt} - \Delta u = 0$ in $\mathbb{R}^{3}$ with $u(x, 0) = f(|x|)/|x|$ ...
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1answer
44 views

Give the example of compact set with infinite countable derived set [on hold]

Can anyone give me an example of compact set of which the derived set is infinitely countable set?? thks in advance, I have no idea about this .
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1answer
34 views

An inequality involving the AM-GM inequality: $| x + \frac1x | \ge 2 $ (for $x<0$).

Suppose $x \neq 0 $, then $| x + \frac{1}{x} | \geq 2 $. I have shown this using the am gm inequality $(a+b)/2 \geq \sqrt{ab} $. In fact, with $a = x^2 $ and $b=1$ works. So, for $x > 0 $ we have ...
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0answers
60 views

Prove √2 exists by Archimedean Axiom [duplicate]

I am trying to prove the existence of the square root of 2. The proof: Let $$S=\{x \in \mathbb{R} ∣x \ge 0, x^2 < 2\}.$$ I understand the proof of LUB, $\alpha$ and so I am at the step where ...
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1answer
20 views

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$.

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$. I'm having difficulty showing the above equalities. I ...
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1answer
23 views

Approximation of characteristic function by mollifiers

I have been asked to show that the Heaviside function $H := \chi_{[0,+ \infty)}$ does not admit weak derivative in $L^1_{loc}(\mathbb{R})$. Here's my reasoning: By definition the weak derivative of ...
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0answers
19 views

Property of nth root

I'm trying to prove the following result: "Let $x,y \geq 0$ be non-negative reals, an let $n,m \geq 1$ be positive integers. If $y=x^{1/n}$ then $y^n=x$." $x^{1/n}:=sup \{y \in \mathbb{R}: y \geq 0, ...
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1answer
14 views

checking definition of bounded linear function involves operator maps between different spaces

Let $H$ and $K$ be two Hilbert spaces. Let $T:K\to H$ be a bounded linear operator. Denote the inner products on $H$ and $K$ by $\langle\cdot,\cdot\rangle_H$, $\langle\cdot,\cdot\rangle_K$. Fix any ...
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0answers
10 views

An elementary inequality in the context of Strictly Convex Function

Suppose: whenever $\epsilon \gt 0$ , define: $\zeta (y, \epsilon) = \frac{\eta (y+\epsilon) - \eta (y)}{\epsilon} - \eta'_{+}(y)$ ; where: $\eta$ is STRICTLY CONVEX CONTINUOUS FUNCTION & ...
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2answers
75 views

Proof that $(a,b)\subset\mathbb{R}$ is not countable. Does it use Axiom of Choice?

I used this proof to show $(a,b)$ is uncountable, but looking at it, I don't really see if it uses AC or not. Until recently I was thinking it does use AC (In the choice of the $a_n,b_n$), now I think ...
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0answers
21 views

About separable spaces

Let $Y$ be the subspace of $B(\mathbb{N},\mathbb{F})$ which consist of all the sequences that tends to zero. Prove that Y is separable. We must show that exists $X\subset Y$ such that $X$ is ...
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26 views

Example of a Riemann integrable sequence of functions such that the the sequence of Riemann integrals diverges but… (see below)?

Is there a sequence $(f_n)$ of Riemann integrable functions such that $\lim f_n(x) = f(x)$ almost everywhere on $[a,b]$ and $\lim\int_a^bf_n$ does not exists in Riemann sense, but it does in Lebesgue ...
3
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1answer
32 views

Largest Triangular Number less than a Given Natural Number

I want to determine the closest Triangular number a particular natural number is. For example, the first 10 triangular numbers are $1,3,6,10,15,21,28,36,45,55$ and thus, the number $57$ can be ...
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2answers
27 views

I am issues with proving the following problem: $f^{-1}(f(A)) ⊃ A$ [duplicate]

I am unsure as to where to start with this problem. The way I read it is that $f^{-1}(f(A)) ⊃ A$ means that $A$ is a subset of the preimage of the image of $A$. But I am unsure.
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1answer
25 views

Discrete analogue of bounded variation

What kind of sequences $(a_n)\subset\mathbb{R}$ are expressible as the difference of two increasing sequences?
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15 views

Average of Convergent Sequences Proof [duplicate]

Show that if ($x_n$) is a convergent sequence, then the sequence given by the averages: $y_n$=($x_1$+$x_2$+...+$x_n$)/n also converges to the same limit. I know that for all $\epsilon$$>$0, ...
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3answers
48 views

Difference between convergence in measure and convergence almost everywhere

This question is an extension of a question asked earlier. Let $(X,\mathcal{M},\mu)$ be a measure space and let $f_{n}: X \to Y$, where $\{f_{n}\}$ is a sequence of functions. The proof wiki ...
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0answers
22 views

Is a compact set in the interior of a cone contained in the intersection of all slightly perturbed cones?

Suppose compact set $S \subseteq R^n$ is in the interior of $x_0+C$, where $C$ denotes a solid convex cone in $R^n$ with the vertex at $0$. I am trying to prove that $\exists r>0$ such that $$S ...
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0answers
15 views

A variation on a problem of Polya and Szego

Among the various propositions on real series and sequences in "Problems and Theorems od Analysis I" Pt. I Chap. 4 by Polya and Szego, I noted n.178 at page 39 which implies what follows. Let ...
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2answers
37 views

Limit of an integral of a continuous real-valued function

If $f:[0,{\infty})\to\mathbb R$ continuous and $\lim_{x\to\infty} f(x)=a$. Show that: $$ \lim_{x\to\infty} \frac1x\int_{0}^{x} f(t)\ \mathsf dt = a. $$ If: $$ \lim_{x\to\infty} \frac1x ...
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1answer
44 views

Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?

I want to know, why $\{e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis of $L^2(\mathbb{T})$, where $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_{\mathbb{T}} ...
2
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1answer
66 views

Finding a general integral

$$ \int\limits_{0}^{1}{\frac{\ln(1+{t}^{a})}{1+t} \;\mathrm{d}t} $$ I have tried many tings but I am just not successful in any of them - Feynman, summation inside integral, Beta function ...
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3answers
46 views

Consider $F(x,y)=f(x+3y,2x-y)$…

If $f: \mathbb{R}^2\rightarrow\mathbb{R}$ where $F(x,y)=f(x+3y,2x-y)$ with $f$ is defferentiable and $\nabla f(0,0)=(4,-3)$ compute the derivate at the origin in the direction of unit vector ...
0
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1answer
22 views

Euclid's Lemma using FToA

I would really appreciate some help understanding the following passage from my Real Analysis text. I have a professor who uses inquiry based learning, which basically means we all stare at each other ...
3
votes
2answers
34 views

Find local extrema of the following function.

Find local extrema of the function $$u(x,y,z)=\sin x \cdot \sin y\cdot \sin z$$ with the condition $$x+y+z=\frac{\pi}{2};\; x,y,z>0$$ Can anyone give me pointers on how to solve this problem? ...
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0answers
14 views

Convergence rate of generalised Fourier series.

Consider a Sturm-Liouville system over an interval $[a,b]$: $$(p(x)y')' + (q(x) + w(x) \lambda) y = 0$$ Induced by this Sturm-Liouville system is a set of special functions that form a complete ...
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1answer
44 views

Problems with understanding analyticity

I have a problem understanding the idea behind Analytic functions. (Please correct me on my terminologies while I state my problem). An analytic function, is a function that has a power series that ...