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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38 views

Prove the existence of the square root of $2$.

I am trying to prove the existence of the square root of $2$. I have some steps with a very vague explanation and I would like to clarify. The proof: Let $$S=\{x\in\mathbb R\mid x\geqslant 0 \text{ ...
0
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0answers
3 views

Proving convergence of a sequence through a polynomial

Let ($x_n$)$\rightarrow$x and let $p(x)$ be a polynomial. (a) Show $p(x_n)$$\rightarrow$$p(x)$. (b) Find an example of a function $f(x)$ and a convergent sequence ($x_n$)$\rightarrow$x where the ...
7
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0answers
47 views

Is there a continuous function from $[0,1]$ to $\mathbb R$ that satisfies

Is there a continuous function $f:[0,1] \to \mathbb R$ such that $f(x) = 0$ uncountably often and, for every $x$ such that $f(x) = 0$, in any neighbourhood of $x$ there are $a$ and $b$ such that $f(a) ...
0
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0answers
19 views

Question about multiplication of Dedekind Cuts?

I need to prove the following: If $r\in\mathbb Q$ and the set $r^*=\{p\in\Bbb Q\mid p<r\}$, then $(rs)^*=r^*s^*$. I already have that $r^*s^*$ is a subset of $(rs)^*$, but I'm stuck trying to ...
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0answers
6 views

Maximum maintains order under limit in $\mathbb{R}^2_{+}$

I'm trying to show that if: $$ (a_{1n},a_{2n})\to (a_1,a_2)\\ (b_{1n},b_{2n})\to (b_1,b_2)\\ max\{a_{1n},a_{2n}\}\geq max\{b_{1n},b_{2n}\},\forall n\in\mathbb{N} $$ Then: $$ max\{a_1,a_2\}\geq ...
1
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1answer
17 views

Convergence Proof Help?

Let ($x_n$) and ($y_n$) be given, and define ($z_n$) to be the "shuffled" sequence ($x_1, y_1, x_2, y_2, x_3, y_3,...x_n, y_n$). Prove that ($z_n$) is convergent if and only if ($x_n$) and ($y_n$) are ...
2
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0answers
22 views

Perturb a piecewise-linear path to make it $C^\infty$

I'm trying to prove that any two points on a path connected smooth manifold can be joined by a smooth path. It becomes easy if I can prove the following: Given a curve $\gamma :\mathbb{R} \to ...
2
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1answer
30 views

If $X$ is compact and $C$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number.

Prove the following statement. If $X$ is compact and $C = \{U_\alpha : \alpha \in A\}$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number. Is my proof correct? Proof: Let $E$ be ...
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0answers
17 views

Limiting value of $L^2$ functions

Let $f\in L^2(\Omega)$, where $\Omega \subset \mathbb{R}^2$ is the unit square $[0,1] \times [0,1]$. Let $x\in \Omega$. Suppose I evaluate $f$ at points from some direction that approach $x$. ...
0
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1answer
18 views

Damped wave equation on $\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$

Let $a \in (0, 1)$ and let $u$ satisfy \begin{align*} u_{tt} - \Delta_{x}u + au_{t} &= 0\\ u(x,0) &= 0\\ u_{t}(x, 0) &= f(x) \end{align*} with $t \geq 0$, $x \in ...
4
votes
1answer
63 views

A triple integral dancing in the unit cube

Straight integration seems pretty tedious and difficult, and I suppose that the symmetry might possibly open some new ways of which I'm not aware. What would your idea be? $$\int_0^1 \int_0^1 ...
3
votes
3answers
169 views

Is $\aleph_0 = \mathbb{N}$?

Some very wise people here have just told me that $\aleph_0 = \mathbb{N}$, i.e. that the cardinality of the set of natural numbers is just the set of natural numbers itself. Is this now the general ...
3
votes
1answer
32 views

Show that $R^2$ cannot be written as a countable union of zero sets of non-trivial polynomials

Problem statement: Show that $R^2$ cannot be written as a countable union of zero sets of non-trivial polynomials. Note that the zero set of a polynomial $p(x,y)$ is $\{(x,y) : p(x,y) = 0$}. My ...
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2answers
34 views

Ternary representation of Cantor set

Associate to each sequence $a=\{\alpha_n\},$ in which $\alpha_n$ is $0$ or $2$, the real number $$x(a)=\sum \limits_{n=1}^{\infty}\frac{\alpha_n}{3^n}.$$ Prove that the set of all $x(a)$ is precisely ...
1
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1answer
16 views

Inequality concerning operator norm

I have difficulty understanding the following statement in PMA by Rudin (aka. "Baby Rudin"): For $A\in L(\mathrm{R}^n,\mathrm{R}^m)$, define the norm $\|A\|$ to be the sup of all numbers $|Ax|$ ...
5
votes
2answers
123 views

Example of the equality of an inequality

This question is related to Daniel Fischer's answer here. Suppose $f$ is a real $C^{1}$ function on $[0, 1]$ such that $f(0) = 0$ and $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$. Then (essentially by ...
2
votes
1answer
31 views

Proving that limit of a sequence is 0 from definitions.

I had this question in a test: Use the definition of limit in order to prove that if $\{a_n\}$ (n goes from 1 to infinity) is a sequence of real numbers such that $\lim_{n\rightarrow \infty} a_n^2 ...
0
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2answers
29 views

Proof that Compact Sets of $R^n$ are measurable

This is from the Stein Shakarchi text, pg 17 - proof that closed sets are measurable. The proof begins by proving that all compact sets $F$ of $R^n$ are measurable. To confirm, does the only reason ...
5
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3answers
130 views

Let $F$ be a sigma algebra such that every element of $F$ is the union of two disjoint nonempty sets also in $F$. Prove that $F$ is uncountable.

I can create a sequence of distinct sets and show that $F$ is countably infinite. I'm looking to create a power set of a countably infinite set, I suppose, but I'm not used to wading so deep into set ...
0
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2answers
45 views

$f(I)\cap g(J)\not=\phi$ for all open interval $I,J$

Let $f,g:\mathbb{R}\rightarrow \mathbb{R}$ and $f(I)\cap g(J)\not=\phi$ for all nonempty open interval $I,J$. Consider $f_1=\chi_\mathbb{Q}$ and $g_1=\chi_\mathbb{Q^c}$, we know that $f_1$ and $g_1$ ...
0
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2answers
14 views

Prove that $ \sum_{k=1}^T t_k f(x_k) \leq B \Rightarrow \min_{ k \in \{1, \ldots, T \} } f(x_k) \leq \frac{ B }{ \sum_{k=1}^T t_k } $

Suppose $f(\cdot)$ is a positive real function, with positive real coefficients $t_k$s, and we know: $$ \sum_{k=1}^T t_k f(x_k) \leq B $$ Can we prove that? $$ \min_{ k \in \{1, \ldots, T \} } f(x_k) ...
0
votes
2answers
43 views

Supremum and infimum of $\left\{(n^2+2n+1)^{\frac{1}{n^2}} \mid n \in\mathbb N \right\}$

Task is to find infimum and supremum of $\left\{(n^2+2n+1)^{\frac{1}{n^2}} \mid n \in\mathbb N \right\}.$ I start from calculating derivative of $ f:\mathbb{R} \rightarrow \mathbb{R}$ where $ ...
3
votes
2answers
212 views

Where does this sequence $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,… converge?

The given sequence is $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,.....and so on. the sequence is increasing so to converge must be bounded above.Now looks like ...
1
vote
2answers
54 views

Dirichlet's Test in convergence [duplicate]

Can Dirichlet's test be applied to establish the convergence of $1 - \frac 1 2 - \frac 1 3 + \frac 1 4 + \frac 1 5 + \frac 1 6 - \dots$ where the number of signs increase by one in each block? Can ...
3
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1answer
460 views

Derivation of poisson kernel for disk of radius $R$ from unit disk

Is there a way to derive poisson kernel for disk of radius $R$ from unit disk?
3
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3answers
44 views

Is $E$ path connected $\implies \overline{E}$ connected?

Let $E\subset \mathbb R^n$ a path connectedness open set. Is $\overline{E}$ connected ? (where $\overline{E}$ is the closure of $E$). I tried to prove that it's true, but I don't get anything, may be ...
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0answers
11 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
2
votes
2answers
54 views

How to prove that in $\{0\} \cup \{1, \frac{1}{2}, \frac{1}{3}, …\}, 0$ is not isolated

(As a subset of $\mathbb{R}$.) I am having trouble proving that $0$ is not an isolated point. If there exists an open ball with radius $\epsilon$ about $0$, I have to find a point of the form ...
2
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5answers
157 views

Proving that $\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}$ [closed]

Proving that $$\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}, \qquad\forall x>\pi$$
0
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2answers
25 views

Upper Bounds and Least Upper Bounds

When defining a least upper bound, how can you have an interval such as (0,1) and have an upper bound of 2?
1
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1answer
42 views

What can be said about the continuous function $f:\mathbb R^{2} \rightarrow \mathbb R$ that has only finitely many $0$'s $?$

$f\colon \mathbb R^{2}\rightarrow \mathbb R$ is a continuous map that assumes $0$ for only finitely many points. Then which one is true A. either $f(x)\le 0$ for all $x$ or $f(x)\ge ...
7
votes
2answers
208 views

Convergence of “alternating” harmonic series where sign is +, --, +++, ----, etc.

Exercise 11 from section 9.3 of Introduction to Real Analysis (Bartle): Can Dirichlet’s Test be applied to establish the convergence of $$ 1 - \dfrac12 - \dfrac13 + \dfrac14 + \dfrac15 + ...
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0answers
10 views

connection between integrablity on the locally compact group and compact subgroup of it

Let G be an locally compact group with Haar measure dx and H is compact subgroup of it with normalize Haar measure dh. If F belong to L^1(G), Is restriction of F to H belong to L^1(H)(i.e. F is ...
0
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1answer
22 views

Is probability mass function (PMF) the “law of X”?

Are they two the same? If not, what's the differences between these two? In continuous case, is PMF also equal to the integration of probability density function?
2
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1answer
20 views

A set $I$ of isolated complex numbers such that $[0,1]\subset\{Re(z):z\in I\}$

Is there a set $I$ of isolated complex numbers, such that $$[0,1]\subset\{Re(z):z\in I\},$$ where $Re(z)$ is the real part of the complex number $z$.
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0answers
41 views
+50

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]\cdot[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
0
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2answers
483 views

An MCQ question on continuity.

Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous bounded function, then : A. $f$ has to be uniform continuous. B. There exists an $x\in\mathbb{R}$ such that $f(x)=x$ C. $f$ cannot be increasing. ...
2
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1answer
24 views

Every ordered field that has the least upper bound property is isomorphic to the real number system.

Okay, so here's a theorem from Rudin: "Every ordered field that has the least upper bound property is isomorphic to the real number system." Here's a definition: "Ordered fields are isomorphic if ...
12
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3answers
546 views

Can we add an uncountable number of positive elements, and can this sum be finite?

Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ...
1
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1answer
35 views

How to understand logically cauchy criterion? [on hold]

Do not provide proof please Just provide a simple ex. to understand it.
0
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2answers
41 views

$f \in C[a,b]$ be such that $\int_c^d f(x)dx=0 , \forall c,d \in [a,b] , c<d$ ; then $f$ is identically zero on $[a,b]$?

Let $f:[a,b] \to \mathbb R$ be a continuous function such that $\int_c^d f(x)dx=0 , \forall c,d \in [a,b] , c<d$ ; then is it true that $f(x)=0 , \forall x \in [a,b]$ ?
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0answers
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Square root algorithm. Rudin PMA ch.3 problem 17

Fix $\alpha>1$. Take $x_1>\sqrt{\alpha}$ and define $$x_{n+1}=\dfrac{\alpha+x_n}{1+x_n}=x_n+\dfrac{\alpha-x_n^2}{1+x_n}.$$ It's easy to check that $\{x_{2n}\}_{n=1}^{\infty}$ is increasing and ...
5
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1answer
49 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
4
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2answers
83 views

Why is $(\mathbb{R}, \mathcal{P}(\mathbb{R}))$ called a measurable space when actually is not?

I get confused when I put the following three notes together: Power set of any set is a $\sigma$-algebra. If $X$ is a set and $\Sigma$ is a $\sigma$-algebra over $X$, then the pair $(X, \Sigma)$ is ...
0
votes
1answer
1k views

Min, Max, Infimum, and Supremum

I have tried to answer the following question but I’m not sure if I’m on the right track… please give me any feedback!! Find the min, max, infimum, and supremum for each of the following. ...
0
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2answers
22 views

The supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ when $x+y=2n$ for some fixed $n\in \mathbb N $

Let $S$ be the set of all tuples $(x,y)$ such that $x+y=2n$ for a fixed $n\in \mathbb N$. Then what is the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ $?$ I substituted $y=2n-x$ ...
10
votes
5answers
2k views

Advice for benefits to directly use analysis textbook to replace calculus

Main purpose: For self-learning performance, neither for exam nor degree courses. Calculus textbook using now[1]: Calculus I, Weinstein&Marsden, UTM, Springer Question Description: I've been ...
0
votes
1answer
43 views

Existence of primitive of a continuous function on an interval (a,b) [on hold]

I like to prove that every continuous function on $(a,b)$ has a primitive but i don't know how to prove it. There is proof that if $f$ is continuous and integrable on finite interval $(a,b)$ then $f$ ...
1
vote
1answer
52 views

Implications of Inner product vs Norm vs Metric Space

Is it true that: -an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality -a norm can be induced by a metric if and only if the metric ...
0
votes
0answers
12 views

norm on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...