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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14 views

If $F=(\frac{-2xy}{(1+x^2)^2+y^2},\frac{1+x^2}{(1+x^2)^2+y^2} )$…

If $F=(\frac{-2xy}{(1+x^2)^2+y^2},\frac{1+x^2}{(1+x^2)^2+y^2} )$ and $a=(0,0),\ b=(x,0),\ c=(x,y)$ compute: $$f(x,y)=\int_{[a,b]}F+\int_{[b,c]}F$$ Someone could explain to me how I could solve that ...
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2answers
15 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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16 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
15 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
47 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 ...
0
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2answers
22 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
16 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
53 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 ...
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4answers
66 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) $ ...
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3answers
78 views

Finding the maximum of a function on $ \Bbb{S}^{8} $.

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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0answers
17 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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0answers
16 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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0answers
13 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 $A, B$ ...
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48 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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0answers
28 views

Show that $\min_K f$ exist.

Let $p\in\mathbb R^n$ and $\|\cdot \|$ the euclidien 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
25 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 two real squences $(a_n)$ and $(b_n)$, define $$ g(x):=\sum_{n=0}^\infty \frac{a_n x^n}{n!}f(b_nx), \quad ...
2
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1answer
27 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
18 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 ...
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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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26 views

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 ...
4
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2answers
58 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.
7
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2answers
95 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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0answers
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
52 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
31 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 ...
2
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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
76 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?
1
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1answer
54 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 $[a,b]$ . ...
1
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1answer
25 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
72 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
28 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 $SU(\mathbb{C}^n)$?
6
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4answers
125 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
118 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
40 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 ...
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2answers
32 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 $$\limsup(a_k + b_k) \leq \limsup a_k + \limsup b_k.$$ ...
2
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3answers
73 views

Limit of sequence of real numbers

Let $\{x_n\}$ be sequence of real numbers such that $\lim_{n\to\infty} x_n=p$. Also if $m\to \infty$ then $n\to \infty$ and converse. How to prove strictly $\lim_{m\to\infty} x_n=\lim_{n\to\infty} ...
10
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4answers
205 views

Find all functions f such that $f(f(x))=f(x)+x$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(f(x))=f(x)+x, \forall x\in\mathbb{R}$. Find all such functions $f$. Clearly, $f$ is an "one-to-one function". I have tried setting ...
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0answers
39 views

Problem 14 from Baby Rudin chapter 3

Let $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1},\quad a_n=s_n-s_{n-1} \quad\text{for} \quad n\geqslant 1$$ Assume ...
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0answers
19 views

Absolute convergence of vector series proof

In Hubbard's multivariable calculus book there is this theorem: If $\sum_{i=1}^{\infty}|\vec a_i|$ converges, then $\sum_{i=1}^{\infty}\vec a_i$ converges. It is said in the book that the ...
4
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0answers
30 views

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 ...
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2answers
16 views

Cantor's Intersection Theorem

If the subsets of the compact space are already non-empty, isn't it obvious that the even the smallest subset is non-empty, and so the intersection is also non-empty because it would be the smallest ...
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1answer
39 views

A question about the Riemann Integrability

I am curious that if there may exist that f(x) is not integrable but we can get a certain value of $\int_a^{\infty}f(x) \,dx$ as the procedure to solve the integrability function?
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1answer
29 views

Mean value formula integrals

Let $f: B(0,R) \rightarrow \mathbb{R}$ be a continuous function. Then I was wondering whether $$\frac{1}{\text{area}(\partial B(0,r))} \int_{\partial B(0,r)} (f(x)-f(0)) dS(x) \rightarrow_{r ...
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0answers
10 views

Density of intersection of sets with boundary condition

I would like to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{n}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) : ...
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0answers
15 views

The image of the interval $(a , b)$ under a lipschitz continuous function

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a lipschitz continuous function. Let $(a , b)$ be an open interval. Is it true that $f((a , b)) = (f(a) , f(b))$?
1
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0answers
30 views

Equivalent Definition of Weak $L^{p}$ (Quasi-) Norm

For a sigma-finite measure space $(X,\Sigma,\mu)$, the weak $L^p$ (hereafter denoted $L^{p,\infty}$) is defined by $$\|f\|_{L^{p,\infty}}:=\sup_{t>0}t\mu(|f|>t)^{1/p}, \qquad (1\leq ...
4
votes
2answers
137 views

The Typewriter Sequence

The typewriter sequence is an example of a sequence which converges to zero in measure but does not converge to zero a.e. Could someone explain why it does not converge to zero a.e.? Note: the ...
0
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2answers
22 views

Simple Question concerning the supremum of functions

Let us say that $x$ and $y$ belong to an interval $I$ . Assume that the function $g$ is bounded on I . is it true that $|g(x)-g(y)|$ is less than or equal to $\sup\{g(x)-g(y)\}$?? Of course this is ...
1
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1answer
35 views

Concavity of function implies convex upper contour

Today I saw a theorem in class that stated the following: $f$ is concave $ \Rightarrow\{z \in \mathbb R^n : f(z) \ge c\}$ is convex. The proof is relatively straight forward and I understand. ...
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2answers
52 views

intuition of mass function of random variable [on hold]

When we are using $P\{X=x\}$ it seems like intuitively there is a function from $T$ (or measure from $\mathcal{B}(T)$) to $[0,1]$. What is the theoretical foundation behind this intuition?