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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5 views

Why a particular type of definition for increasing sequence is chosen?

I am asking the following question in the context of first-level analysis course. A increasing sequence is generally defined as: A sequence is increasing if and only if $a_{n+1} \gt= a_n $ for all n. ...
-1
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3answers
52 views

Confused about transcendental numbers

I'm little confused about the type of numbers that had been known, for example, consider a polynomial equation with rational and irrational coefficients of a degree p-prime number that is greater than ...
0
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4answers
63 views

Is an non-invertable matrix an linear operator?

I am under the impression that any matrix can be called a linear operator, even if the matrix does not have an inverse. Is it true? There are many properties a linear operator enjoys; do all matrices ...
1
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2answers
49 views

Show that this series is divergent using the comparison test.

Let $\langle a_n\rangle$ be a sequence of positive numbers. Consider $\sum_{n=1}^\infty{Y_n}=\sum_{n=1}^\infty{\left(\frac{a_n}{n}+\frac{n}{a_n^2}+\frac{a_n}{n^3}\right)}$. Show this diverges using ...
-1
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0answers
46 views

An inequality with a real number represented by a Dedekind cut [on hold]

Theorem 3: Let $r$ be a real number $(A,B)$ (meaning a Dedekind cut). Then $r \ge a$ for all $a$ in $A$, and $r < b$ for all $b$ in $B$. If $r$ is irrational, then $r > a$ for all $a$ in $A$. ...
2
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0answers
9 views

$L_2((-2,2))$ function that has $L_1((-1,1))$ discrete derivative but not derivative

I am trying to find an example of a function $u\in L_2((-2,2))$ such that $||\delta_h(u)||_{L_1((-1,1))}$ is uniformly bounded in $0<|h|<1/2$ but $u'$ is not in $L_1((-1,1))$. Where ...
4
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1answer
64 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|]$$ This question is a re-posting of An expectation inequality. I can ...
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0answers
17 views

Limit Theorems for the ArgMax of Function?

I have a continuous, infinitely differentiable function $f_n(x)$. I know the functional form of the function and any of its derivatives. I know $f_n(x) \rightarrow 0$ uniformly in $x$ as $n ...
1
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1answer
32 views

“Inverse” of nondecreasing, right-continuous function?

Suppose $F : \mathbb{R} \to \mathbb{R}$ is a nondecreasing and right-continuous function. Define $G : [\inf F,\sup F] \to \overline{\mathbb{R}}$ by $G(p)=\inf \{ x : F(x) \geq p \}$, with the ...
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1answer
19 views

Boundary preserving map

Let $K\subseteq\mathbb{R}^2$ be a compact set. Is it true that for a continuous map $p:K\to\mathbb{R}^2$ we have: $p(\partial K)=\partial p(K)$? Are there any generalizations? P.S. Note that ...
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2answers
45 views

Solve $1/x^2 = \sum_{i=1}^{n} \frac{1}{x+a_i}$ over $x>0$

Does the equation $\frac{1}{x^2} = \sum_{i=1}^{n} \frac{1}{x+a_i}, \qquad a_i > 0, \quad i=1, \ldots, n$ always admits one and only one solution $x^* > 0$? If yes, what is the most elegant ...
4
votes
2answers
39 views

dominated convergence for functions $\mathbb R^n\to\mathbb R^m$?

I do know the dominated convergence theorem for functions $f:\mathbb R^n\to\mathbb R$. Now let $U\subset\mathbb R^n$ and $f: U\to\mathbb R^m$. Is there any dominated convergence theorem for ...
2
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0answers
25 views

Closure of Schwartz spaces in $L^2$-spaces

I am stuck with the following question and I am hoping that someone could help me with it: Let $D$ be the line $y=0$ in $\mathbb{R}^2$. I define the Schwartz space $\mathcal{S}(\mathbb{R}^2\backslash ...
1
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1answer
35 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
-3
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1answer
29 views

A problem of the limit of a serie [on hold]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Calculate the following limit: Thanks!
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1answer
12 views

Prove $f\in \mathscr{R}(\alpha)$ and $\int_a^b f\ d\alpha = f(s)$ with the following conditions.

If $a<s<b$, $f$ is bounded on $[a,b]$, $f$ is continuous at $s$, and $\alpha(x)=I(x-s)$, then prove that: $$f\in \mathscr{R}(\alpha)$$ and $$\int_a^b f\ d\alpha = f(s)$$ $I$ is a unit step ...
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0answers
58 views

A not very easy problem… [on hold]

I leave a challenge, a derivative problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - \alpha x \sin x = \mathcal{O}(x^4), \text{ as } x\to 0 $$
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0answers
14 views

Compactness of a convex collection

Given $\epsilon\in(0,1)$, suppose we have collection $\mathscr{C}(\epsilon)$ of multilinear polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ is in range $(-\epsilon,\epsilon)$ on $S_0$ while ...
0
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0answers
26 views

How is this convex set compact as well?

Given $\epsilon\in(0,1)$, supposing we have a collection $\mathscr{C}(\epsilon)$ of polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ takes on value $0$ on $S_0$ while being in range ...
0
votes
1answer
30 views

Proposition on limsup

Given sets (or events) $A_1, A_2, A_3, ...$ and function $f: \mathbb{N} \to \mathbb{N}$, show that $\limsup A_{f(n)} \subseteq \limsup A_n \Leftrightarrow f(n) \to \infty$ This is what @Did told me ...
3
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2answers
42 views

if $f:(a,b) \rightarrow (c,d)$ is a differentiable surjection and $f'(x)$ is never zero then $f$ is a homeomorphism

There is a theorem in our analysis book which says: if $f:(a,b) \rightarrow (c,d)$ is a diffrentiable surjection and $f'(x)$ is never zero then $f$ is a homeomorphism. This is the proof of the book: ...
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1answer
28 views

Show that if the set A is bounded then the set Bϵ(A) is bounded.

Let $A$ be a nonempty closed subset of $\mathbb{R}^{n}$. For $\epsilon>0$, $B_{\epsilon}(A)= \left \{ x\in \mathbb{R}: d(x,A)<\epsilon) \right \}$. a) Show that if the set $A$ is bounded then ...
2
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1answer
43 views

Prove that all the five sequences converge to the same point $P \in \mathbb{R}^3$.

Let five sequences $A_n, B_n, C_n, D_n, E_n \in \mathbb{R}^3$ be constructed as follows: $A_0, B_0, C_0, D_0$ and $E_0$ are some given points of the space and $A_{n+1}, B_{n+1}, C_{n+1}, D_{n+1}, ...
0
votes
2answers
43 views

Closure of $\Bbb R$ [on hold]

Is the closure of $\Bbb R$ equal to $\Bbb R$ itself or the extended real numbers $\bar{\Bbb R}$? Thanks for any comment.
1
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1answer
26 views

Intuition of weak star convergence.

Given $\Omega=(0,1)$, consider the following sequence $$ v_j(x)\colon=\begin{cases} \;a &\text{if }jx-\lfloor jx \rfloor\le\theta\\ \;b &\text{otherwise} \end{cases} $$ where ...
-3
votes
1answer
36 views

The integer part of $x+1$ is the integer part of $x$ plus $1$ [on hold]

How do you solve the proof: If $x$ is a real number, then: $[x+1] = [x] + 1$. For my proof, I tried to describe the interior of the argument inside the parentheses, but I was unsuccessful. Please ...
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2answers
53 views

Prove $\lim_{x\to+\infty}\frac{x^3}{|x|} = +\infty$

We want to prove $\lim_{x\to +\infty}\frac{x^3}{|x|} = \infty$. I don't think the usual "$\varepsilon-\delta$ definition would work since we are dealing with $\infty$. How else could I approach this ...
0
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0answers
18 views

Proving Euler's theorem for homogeneous functions.

This problem is from Apostol's Mathematical Analysis. Let $f$ be defined on an open set $S$ in $R^n$. We say that $f$ is homogeneous of degree $p$ over $S$ if $f(\lambda x)=\lambda ^p f(x)$ for every ...
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0answers
27 views

A Measure Problem on Stein's Real Analysis

I'm considering problem 5 on Stein's real analysis chapter 6 $X$ is a metric space, for any positive linear functional $l$ on $C_0 (X)$ which are the continuous functional on $X$ supported in some ...
2
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0answers
34 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
0
votes
2answers
28 views

Find the limit points and exterior points of the following

Let $X=\mathbb R$, with the usual metric on $\mathbb R$ and $A=((0,1)\cap \mathbb Q)\cup$ {$2,3$}. Find the limit points of $A$, exterior points of $A$, $A^o$, $\overline A$ and $\partial A$. Can ...
1
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2answers
38 views

Does $(f_n)$ converge pointwise/uniformly on $I$?

Does $(f_n)$ converge pointwise/uniformly on $I$ if $$f_n(x) = \frac{x^n}{1+x^n} ~~~~~~ I=[0,1]$$ My attempt: If $x \in [0,1): \displaystyle \lim_{n \to \infty}f_n(x) = 0$ If $x=1: ...
0
votes
1answer
30 views

Is there a relation between Ill-posed problems and Eigenvectors.

One can easily explain an ill-posed problem with an equation AX=b. The following link is an good example: http://www.encyclopediaofmath.org/index.php/Ill-posed_problems 1) Can there be a class of ...
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0answers
36 views

Show that subspace metric induces subspace topology [on hold]

Let $(X,d)$ be a metric space, let $\tau$ be the topology on $X$ induced by $d$ and $A \subset X$. Define $d_A: A \times A \to \mathbb R$ as $d_A(a,b)=d(a,b) \forall a,b \in A$ . Show that $d_A$ ...
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2answers
60 views

Is $[-1,1]$ complete under the Euclidean metric? [on hold]

Is it true that the interval $[-1,1]$ is complete under the Euclidean metric?
1
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1answer
31 views

Show that $\tau_A$ is a topology on $A$

Let $(X,\tau)$ be a topological space and $A \subset X$. Let $\tau_A$={$A \cap U: U \in \tau$}. Show that $\tau_A$ is a topology on $A$. I know that I need to prove three properties to prove ...
2
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1answer
53 views

Please check my demonstration of de l'hopital's rule

I have demostrate the de l'hopital theorem but in some steps I'm not 100% sure; The theorem I demostrate is for: $\lim_{x\rightarrow a+} \frac{f'(x)}{g'(x)}=L \implies\lim_{x\rightarrow a+} ...
2
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1answer
31 views

Open-Set Correspondence $\implies$ Continuity

I would like to show the following implication. Let $f$ be a function from $\mathbb{R}^n$ to $\mathbb{R}^m$. If $f^{-1}(U)\subset\mathbb{R}^n$ is open for every open $U\subset\mathbb{R}^m$, then ...
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0answers
31 views

Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$

Let $(x_n)$ be a sequence in $X$ and $x \in X$. $(X,d)$ is a metric space and $f: X \times X\to $ is a bijection. Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$. ...
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0answers
22 views

If a bounded sequence is equicontinuous, it has a uniformly convergent subsequence

I am currently having some difficulty with problem 2.7.8 in Introduction to Topology by Theodore Gamelin and Robert Greene. The problem goes as follows A family F of real-valued functions on a ...
0
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3answers
53 views

Show $f$ is not continuous on $\mathbb{R}$

Show that the function $$f(x) \begin{cases} 1 & \text{if } x\in \mathbb{Q} \\ 0 & \text{if } x \not\in\mathbb{Q}\end{cases}$$ is not continuous anywhere in $\mathbb{R}$. Give reason(s) for ...
1
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3answers
46 views

Explicit form of this series expansion?

I am considering the following series expansion: $$f(k):=\sum_{n\geq 1} e^{-k n^2}$$ with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at ...
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2answers
31 views

Show that $d_f$ is a metric on $X$ [on hold]

Let $(X,d)$ be a metric space, and let $f: X \to X$ be a bijection. Define $$d_f: X \times X \to \mathbb R $$ as $d_f(x,y)=d(f(x),f(y))$ $\forall x,y \in X$ Show that $d_f$ is ...
1
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3answers
43 views

Show $\exists a >0$ such that $f(x) \ge a$

Let $f:[0,1] \to \mathbb{R}$ be continuous and suppose $f(x)>0$ for all $x \in [0,1]$. Show that there exists a number $a>0$ such that $f(x) \ge a$ for all $x \in [0,1]$. I have a feeling ...
3
votes
1answer
28 views

Two versions of the Inverse Function Theorem.

I first learned about the Inverse Function Theorem for $C^1$ functions in Rudin's Principles of Mathematical Analysis in the following form. Inverse Function Theorem ($C^1$ version): Suppose $E$ ...
3
votes
1answer
132 views

Counterexample for Interchange of Limits in integration

If $f_n$ converges to $f$ uniformly in $\mathbb{R}$, then \begin{equation*} \lim_{n\to\infty}\int_a^b f_n(x)\,dx =\int_a^b f(x)\,dx \end{equation*} but it's not true in general that ...
0
votes
2answers
49 views

What kind of set-theory is sufficient to understand mathematical analysis?(book recommendation))

I am looking for books with set theory and logic that is sufficient to understand mathematical analysis. I guess another question might be if there even exists such a book. There are basically two ...
0
votes
1answer
37 views

Differentiability question using the definition

Could anyone please help me with where to start with this question? (c)
2
votes
0answers
34 views

Having difficulty calculating a derivative using the definition and limit question

My first question is how do we justify that the limit of $\sin (x)\sin \frac{1}{x}$ as $x$ tends to $0$ is $0$? Would we say $-\sin(x)\leq \sin (x)\sin (\frac{1}{x})\leq \sin (x)$ and then use the ...
0
votes
0answers
6 views

Deriving information about asymptotics from finitness of a limit

Let $f_1,f_2:\mathbb{R}\setminus\{0\}\to \mathbb{R^+}$ be two $C^1$ functions and $\alpha:\mathbb{R}\setminus \{0\}\to \mathbb{R}$ be a function from a Zygmund class (in particular it is Holder for ...