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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If $\phi$ vanishes outside of $|x| < 2$, $\phi = 1$ where $|x| < 1$, prove/disprove $f * \phi$ is in $L^1$/$L^2$

Problem Statement: Let $\phi$ be a positive, smooth function. Suppose $\phi$ vanishes outside a compact subset of $\{x : |x| < 2\}$ and satisfies $\phi(x) = 1$ if $|x| < 1$. Let $f$ be a ...
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19 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 ...
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10 views

Differentiability of parameter-dependent integrals when derivative exists only almost everywhere

This unanswered question asked in 2013 Differentiation under the Integral Sign (let's call this Q-zero) seems to be taken from this (or pdf ver.). The result on differentiation under the integral ...
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1answer
28 views

linear algebra (norm)

Can someone explain to me the following definition - $\|T\|$ := $ \sup \{\|T(v)\| : v \in \mathbb{R}^n, \|v\| = 1\}$ where $T$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ and ...
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2answers
49 views

Analysis in $R^n$

I have to justify whether this statement is true or false - If a linear transformation(matrix) $||T||$ is non-invertible then $||T||$ = $0$. $||T||$ is the norm. Justification - $||T||$ is = $0$ iff ...
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0answers
9 views

Geometric Understanding of Dyadic Squares

According to Pugh's Real Mathematical Analysis: A rational number $\frac{p}{q}$ is dyadic is $q$ is a power of $2, q = 2^k$ for some non-negative integer $k$. A dyadic interval is $[a,b]$ where ...
4
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2answers
48 views

How to simplify $ \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx $ using Green's indentity?

Let $\varphi\in C_c^\infty(\Bbb{R^2})$ (infinitely differentiable functions with compact support) and consider $$ I=\int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx, $$ the existence of which is ...
4
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2answers
67 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 ...
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2answers
311 views

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

Can we add an uncountable number of elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. Any help ...
5
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1answer
24 views

Proving that and how $ \frac{1}{n}\sum\limits_{k\le n}\lfloor n/p_k\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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1answer
38 views

A continuous bounded function from $\mathbb R$ to $\mathbb R$ can be increasing or not?

Let $f:\mathbb R \rightarrow \mathbb R$ be a continuous and bounded function , then $a$) $f$ has a fixed point. $b$) $f$ cannot be increasing $c$) $\lim_{x\rightarrow \infty} f(x)$ exists. ...
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0answers
45 views

Determining the interior of $([-1, 1]\times[-1, 1])\setminus \{ y \in \mathbb{R}^2 : d((0, 0), y) < 0.25 \} \subseteq \mathbb{R}^2$

Let $M = (\mathbb{R}^2, d_e)$ be the metric space, with $d_e$ the Euclidean metric. Let $C \subseteq \mathbb{R}^2$ be defined by $$C = ([-1, 1]\times[-1, 1]) \setminus \{ y \in \mathbb{R}^2 : d((0, ...
4
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2answers
110 views

A singular Gronwall inequality

Let $f : [0,T] \to R^+$ be a continuous function such that $f(0)=0 $ and : $$ f(t)\le C\int_0^t s^{-1}f(s) ds,\; \forall t\in [0,T] $$ for some constant $C>0.$ Is it true that $f(t)=0,\; \forall ...
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1answer
43 views

Every convex set in $\mathbb R^n$ has a countable and dense subset?

Assume the space is Euclidean space. Why every convex set has a countable and dense subset? How about in metric space? Any ideas or references? It is used in process of proving Debreu's Theorem in ...
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2answers
20 views

Finding a minimum of a function, measuring the sum of the squares of distance from some points of the $\mathbb{R}^n$

Given are a finite number of points $a_1, ..., a_m \in \mathbb{R}^n$. Consider the sum of the squares of distance: $$f(x) = \sum_{k=1}^m ||x-a_k||^2, x \in \mathbb{R}^n$$ with $||.||$ being the ...
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0answers
19 views

Finding a function satisfying a certain inequality

This is a continuation of this post where I tried to find a function $f(n)$ that would satisfy the induction step of an inductive argument and it was shown that such function does not exist. Trying ...
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1answer
25 views

Concavity of distance function in $\mathbb{R}^n$ or determinant of $(x^T \cdot x)$

I would like to compute the concavity of the distance function in $\mathbb{R}^n$. Let $ f(x) =- \Vert x \Vert $ in $\mathbb{R}^n$. Then $\nabla_xf=- \frac{x}{\Vert x \Vert}$. And ...
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0answers
29 views

Exercise 3.32 from Real Analysis of Folland

Can someone give me some hint on how to solve this problem? Thanks a lot If $F_1, F_2, ..., F \in NBV$ and $F_j \rightarrow F$ pointwise, then $T_F \le \liminf T_{F_j}$ Here, NBV is the ...
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1answer
17 views

Limit of a Monotonic Increasing and Non-Bounded Function

I have made a solution for the following question and I'm wondering if it's correct. I think that something is missing here. Can you help me complete the solution? Let $f$ be a function. The ...
2
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1answer
53 views

Algorithm for computing square roots.

Fix a positive number $\alpha$. Choose $x_1>\sqrt{\alpha}$ and define $x_2, x_3, x_4, \dots$ by the recursion formula $$x_{n+1}=\frac{1}{2}\left(x_n+\frac{\alpha}{x_n}\right).$$It's easy to check ...
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2answers
31 views

Weak convergency vs strong convergency in Hilbert space

Let $\mathcal{H}$ be an Hilbert space and let $(x_n)_n \subset \mathcal{H}$ be a sequence s.t. $$ x_n \rightharpoonup x ~~~,~~~ \| x_n \| \to \|x\| $$ We want to show that $ x_n \to x $. Now, I ...
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1answer
21 views

Problem: conservative and not conservative $F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right)$

I don't know how I can solve this problem: Consider $$F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right).$$ Proving that $F$ is not conservative in $\mathbb{R}^2-(0,0)$ but is conservative ...
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3answers
63 views

How to prove that this function is integrable on $[0,1]$

Here I tried to find two step functions, one of them is less than $f$ on $[0,1]$ whereas one of them is greater than $f$ on the same closed interval, to prove this function is Riemann-integrable on ...
0
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4answers
72 views

Convergence or divergence of the series $\sum\limits_{n = 1}^{\infty} \sin(\pi/n)$

Let $ u_{n} = \sin \! \left( \dfrac{\pi}{n} \right) $, where $ n \in \Bbb{N} $, and consider the series $ \displaystyle \sum_{n = 1}^{\infty} u_{n} $. Which of the following is/are true? (a) $ ...
5
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3answers
110 views

Finding the maximum of a function on $ \Bbb{S}^{7} $.

I'm trying to find the maximum of the function $$2 a^2 h+\sqrt{3} a d f+\sqrt{3} a e g+2 b^2 h-\sqrt{3} b d g+\sqrt{3} b e f\\+2 c^2 h+\sqrt{3} c d^2+\sqrt{3} c e^2-\sqrt{3} c f^2-\sqrt{3} c ...
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24 views

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

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 $f$ ...
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1answer
27 views

A question about a generated $\sigma$-algebra of a family set

Wikipedia's definition of Family of sets: In set theory, a collection $F$ of subsets of a given set $S$ is called a family of subsets of $S$, or a family of sets over $S$. So suppose $Ω$ is ...
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1answer
14 views

Classify the growth of functions and find a more general growth function

The following function $f(t,x):[0,T]\times R\mapsto R$ such that $\int^T_0|f(t,0)|^2 d t<\infty$, where $0<T<\infty$. If $f(t,x)$ satisfies $|f(t,x)|\leq Ax+B$ for each $x\in R$ and $A, B$ ...
3
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50 views

$f$ is continuous, $f : X \to X$, $X$ compact, and $f$ has an $\epsilon$-fixed point for each $\epsilon > 0$. Show $f$ has a fixed point.

Problem: Let $f : X \to X$ be a map from a metric space to itself. A point $z \in X$ is a fixed point of $f$ if $f(z) = z$. Let $\epsilon > 0$. A point $w \in X$ is an $\epsilon$-fixed point of $f$ ...
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1answer
33 views

Every closed subset of $\mathbb R^n$ has a point that minimizes the distance to a given point $p\in\mathbb R^n$

Let $p\in\mathbb R^n$ and $\|\cdot \|$ the Euclidian norm. Show that if $K\subset \mathbb R^n$ is a close set, then $$\exists a\in K: \forall x\in K, \|a-p\|\leq \|x-p\|.$$ Since $\|x-p\|\geq 0$, ...
2
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1answer
35 views

When does $\sum_{n=0}^\infty \frac{a_n x_n}{n!}f(b_nx)$ converge for $f\in C_c^\infty(\Bbb{R})$?

Let $f\in C_c^\infty(\Bbb{R})$ be such that $f(x)=1$ for $x\in (-1,1)$. Given a real sequence $(a_n)$, define $$ g(x):=\sum_{n=0}^\infty \frac{a_n x^n}{n!}f(b_nx), \quad x\in\Bbb{R} $$ where ...
2
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1answer
29 views

Limit of a recurrence

I was given the following exercise as homework: find the limit of $b_{n+1} = \sqrt{2 + b_n}$, $b_1 = \sqrt{2}$, with a hint that $b_n < 2 \forall n \in \mathbb{N}$. I have proven that $b_n$ is ...
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1answer
19 views

Does weak-$\ast$ convergence with an exponential rate imply convergence of measures of sets with the same rate?

Assume that $\mu_n \to \mu$ in the weak-$\ast$ topology with the following rate for any compactly supported continuous function $f$: $$|\mu_n(f) - \mu(f)| \leq C_f e^{-n}.$$ Can we replace $f$ with ...
1
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1answer
17 views

uniform continuity

Let $F(s,y)$ be uniformly continuous in $[a,b] \times B$, where $B \subset R^n$ is a closed subset. Assume $x_k \rightarrow x$ in $C[a,b]$ with $x_k(t) \in B$ and prove $$\int_a^b F(s,x_k(s)) ds ...
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Understanding the set structure of probability theory [on hold]

Since events have their own probabilities and outcomes have their own probabilities. Why don't we just consider only one of events or outcomes directly? What's the motivation to have this set-point ...
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2answers
60 views

If $\sum_{m,n}a_{mn}x^m(1-x)^n\equiv 0$, can we conclude $a_{mn}=0$?

Assume $\{a_{mn}\}$ are some real numbers between -1 and 1. If we know $$\sum_{m,n}a_{mn}x^m(1-x)^n\equiv0\quad\forall x\in(0,1),$$ can we conclude that $a_{mn}=0$ for all $m,n\geq 0?$ Thanks.
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100 views

Real analytic functions

I'm writing because I don't know the usefulness of real analytic functions. I mean, I know that analyticity is something more respect differentiable ($C^\infty$ function), but I don't have in mind a ...
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20 views

How to obtain accumulated counts of past events by time $t$?

Given $f: [0, \infty) \to \{0,1\}$, $f(t)$ represents whether there is an event occurring at time $t$. How can we obtain $g: [0,\infty) \to \mathbb{N}_0$ so that $g(t)$ represents the number of ...
4
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3answers
54 views

When can a set have an upper bound but no least upper bound?

So I'm taking real analysis and have noted that one of the benefits of the Dedekind cut is that 'if one of the sets made has an upper bound it also has a least upper bound'. I don't understand how a ...
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3answers
33 views

Uniform convergence polynomial -Stone Weierstrass

A consequence of the Stone Weierstrass theorem, is that for any continuous real function $f$ on a closed bounded interval in $\mathbb{R}$, we can find a sequence of real polynomials $f_{n}$ converging ...
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2answers
34 views

Injectivity of the function $x||x||$ on $\mathbb R^n$

Let , $f:\mathbb R^n\to \mathbb R^n$ be a function defined by $f(x)=x||x||^2$ for $x\in \mathbb R^n$. Then , which are correct ? (A) $f$ is one-one. (B) $f$ has an inverse. Here $f$ is not a ...
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5answers
80 views

Does the limit $\lim\limits_{x\to0}\left(\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}\right)$ exist?

Does the limit: $$\lim\limits_{x\to0}\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}$$ exist?
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1answer
58 views

If $f$ is differentiable and $f'$ is bounded then relation between upper sum , lower sum and the integral

Let , $f:\mathbb R\to \mathbb R$ be a differentiable function such that $f'$ is bounded. Given a closed and bounded interval $[a,b]$ and partition $P=\{a=a_0<a_1<\cdots <a_n=b\}$ of $[a,b]$ . ...
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1answer
26 views

Limit Help: $\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}$

I feel dumb for asking this, but I couldn't quite show that this limit is 0 (which I think is correct) whenever $a>0$: $$\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}.$$ I tried using L'Hospital's ...
4
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0answers
77 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...
2
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1answer
29 views

$U(\mathbb{C}^n)$, $SU(\mathbb{C}^n)$ connected subsets of $M_n(\mathbb{C})$?

As the title suggests, is $U(\mathbb{C}^n)$ a connected subset of $M_n(\mathbb{C})$? How about $SU(\mathbb{C}^n)$?
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4answers
131 views

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$?

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$? Obviously we may as well assume all the subsets have measure $0$. If I didn't specify the subsets were ...
0
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2answers
119 views

How do mathematicians find the underlying idea?

While reading through the lecture notes here (http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week2.pdf , page 22, last paragraph), I came across the following " Thus there must be some ...
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1answer
43 views

A question about 2.1 Proposition on Folland's Real Analysis

Definition of measurable space: If $X$ is a set and $\mathcal{M} \subset \mathcal{P}(X)$(Power set of $X$) is a $\sigma$-algebra, $(X, \mathcal{M})$ is called a measurable space and the ...
0
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2answers
36 views

Prove the following properties of sequence

Define $$L = \limsup_{k \rightarrow \infty}a_k =\inf_j(\sup_{k\geq j}a_k).$$ Prove that if $(a_k)$ and $(b_k)$ are sequence of real numbers then $$\limsup(a_k + b_k) \leq \limsup a_k + \limsup b_k.$$ ...