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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19 views

Limit of Convergent Sequence Property Proof Help

I have a question about this property: Let $\lim\limits_{n\to\infty} a_n = a$, then $\lim\limits_{n\to\infty}(ca_n) = ca$ for all $c \in \mathbb R$ If we consider when $c$ doesnt equal $0$, my ...
0
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1answer
57 views

Prove that $f'(c ) = 0$

Let $f: (a,b) \to \mathbb{R}$ be a function defined on $(a,b)$. Let $c \in (a,b)$ be a local maximum and $f'( c)$ exists. Prove that $f'(c ) = 0$ Sometihng I have thought so far: For some $\delta ...
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0answers
26 views

How to prove it and how to solve it

Tomorrow I will begin my studies, real analysis, however I have some difficulties in making statements so I thought before starting the study in real analysis, learn how to do demonstrations properly. ...
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3answers
28 views

function bounded by an exponential has a bounded derivative?

here's the question. I want to be sure of that. Let $v:[0,\infty) \rightarrow \mathbb{R}_+$ a positive function satisfying $$\forall t \ge 0,\qquad v(t)\le kv(0) e^{-c t}$$ for some positive constants ...
0
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2answers
17 views

exponential boundedness of components given exponential boundedness of the norm

Let $v:[0,\infty)\rightarrow \mathbb{R}^n$ be a function such that $\forall t\ge 0$, $v_i(t)\ge 0$ and $$ ||v(t)||\le \beta ||v(0)||e^{-at}, t\ge 0$$ with $\beta,a>0$ can I conclude that for all ...
0
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1answer
32 views

Existence of such a function

I am supposed to construct a function $f \in C_c^1((-\frac{3R}{4},\frac{3R}{4}))$ such that $f|_{(-\frac{R}{2},\frac{R}{2})}=1$ and $|f'(x)| \le \frac{4}{R}$ for almost all $x \in (-R,R)?$ I ...
2
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4answers
17 views

Difference of consecutive pairs of sequence terms tends to $0$

This seems an elementary problem, but I don't know of any reference to it in the literature. Consider the sequence $(a_n)_{n=1}^\infty$ of real numbers. Suppose ...
0
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0answers
10 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
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0answers
15 views

Image of a Sufficiently Small Open Set Under a Constant Rank Map Cannot Twist Too Much

$\newcommand{\R}{\mathbf R} \newcommand{\norm}[1]{\|#1\|} \DeclareMathOperator{\ball}{B}$ I am trying to prove the following rather visually obvious "fact", as the title describes, preferably in a ...
0
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1answer
28 views

which hypothesis for boundedness of this function

Let $v:[0,\infty)\rightarrow \mathbb{R}_+$ be a positive function such that $$\exists T,q>0\,\,s.t.\,\, \forall t\in[0,\infty),\,\,\int_t^{t+T} v(\tau) d\tau \le q$$ I'm looking for the "less ...
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0answers
19 views

Limit of a recursive sequence containing log [on hold]

Let $\alpha$ be a real number. Consider the following recursive formula: $a_1=1$ and $$a_n=1-\alpha . \sum_{i=1}^{n-1}{a_i\over{i.\log(n-i+1)}} \: \: \: \:for\:\:n\ge2$$ Note that the logarithm is ...
4
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0answers
43 views

The quadratic and cubic versions of a tough intregral

In this post, Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$, it's proved that $$I_1=\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log ...
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0answers
12 views

Metric for connected path space.

I'm trying to prove the next function is a metric for the space of connected paths $T_{x,y}(X)$ where $x,y\in X\subset\mathbb{R}^{n}:$ $$d(x,y)=\inf\{L(\sigma):\sigma\in T_{x,y}(X)\},$$ where ...
0
votes
2answers
16 views

limit points of subset of real numbers

Let $$A=\{ \frac{\sqrt{m} -\sqrt{n}}{\sqrt{m}+\sqrt{n}} | m,n\in \Bbb{N} \}$$ I think that we must find sequence of $A$ and find limit of sequence,let $a_m =\frac{\sqrt{ k ^2 m^2} -\sqrt{ ...
1
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3answers
39 views

Finding $\lim\limits_{n\to \infty}({1\over n+1}+{1\over n+2}+…+{1\over n+n})$ using integrals [duplicate]

Finding $\lim\limits_{n\to \infty}\left({1\over n+1}+{1\over n+2}+\dots+{1\over n+n}\right)$. I tried many things but it would work out. I am now studying calculus 2 (In my country the first calculus ...
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0answers
29 views

Question about weak derivatives

I have a question about weak derivatives. Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some open set $\emptyset \neq U \in \mathbb{R}^{n}$ We often say that $v$ is the ...
2
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0answers
34 views

Ball with euclidean metric - mistake in book?

In $\mathbb{R}^2$, the ball with euclidean metric $d_{l_2}$ is defined, in terence tao's analysis vol II, as: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
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0answers
25 views

Existence theorems depending on compactness of unit ball?

I can only think of that a semi-continous function attain it's maximum on compact sets. What other existance themorems depend on compactness of unit ball? Which cases are we able to maintain and which ...
2
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0answers
29 views

Total variation measure vs. total variation function

Let $a, b \in \mathbb{R}$ with $a < b$ and define the compact interval $I := [a, b]$. Let $g, h : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous on $I$ and constant on ...
1
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0answers
26 views

Proving one expressions is greater than the other using limits?

In general, is it sufficient to show that one of them increases faster than the other? $$1-P_{k,1}< or > (1-P_{k,2})(M+B(1-p))/(M+B))$$ where $P_{k,1}$ and $P_{k,2}$ are decreasing with M. ...
3
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1answer
55 views

Asymptotic behaviour of $\int_0^1 g(x)\exp(-nx)dx$ as $n\rightarrow\infty$

Let $g:(0,1]\rightarrow\mathbb{R}_+$ be an invertible monotonically non-increasing function that integrates to $1$ and has $g(1)=0$, $g(0)=\infty$; eg. $g(x)=x^{-1/2}-1$ or $g(x)=\ln(1/x)$. I believe ...
1
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1answer
38 views

Analysis for Engineering : Practical Applications

I don't know much more about Analysis than what I've read about it on Wikipedia, although I have just begun reading Introduction to Calculus and Analysis I, by Richard Courant. My understanding is ...
1
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2answers
39 views

Finding limit points for these sets

Here's my resoning for finding limit points for some sets. Could you guys read it and see if it's all good? <3 $$\{(x,y)\mid \ x^2+y^2<1\}$$ For this set, its kinda simple to see that every ...
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0answers
27 views

Prove a certain function is discontinuity type I and integrable. [on hold]

Let $x\in\mathbb{R}$ and let $m(x)$ be the unique integer minimizing the value $|x-m(x)|$, for $x \neq n/2$ for $n$ odd. Let $$ (x):=\begin{cases} x-m(x) & \mbox{ if } x\neq n/2, n \mbox{ odd ...
2
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3answers
69 views

Show that $f$ is bounded.

Let $-\infty<a<b<\infty$. Suppose $f$ is continuous on $[a,b]$. Show that $f$ is bounded on $[a,b].$ We are supposed to use intermediate value theorem for this problem. But, I don't ...
0
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0answers
49 views

Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$ [on hold]

Let $f\in C^{\infty}(Ω)$ for some open set $Ω \subset R^n$ that contains $0$. Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$. I found this problem in a ...
3
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1answer
44 views

A negative third derivative implies a positive first derivative at a point.

Let $f$ be three times differentiable on $\mathbb{R}$ with $f(0)=f(1)$, and for all $x\in[0,1]$, $f'''(x)<0$. Prove that $f'(\frac{1}{2})>0$ I actually have a proof of this question using ...
1
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0answers
27 views

Functions linearly independent and linearly independent gradients?

Let $F_1,...,F_n: \mathbb{R}^n \rightarrow \mathbb{R}$ be a set of $C^{1}$ functions. Is it true that they are linearly independent on a joint level set $\Omega:= \{ p \in \mathbb{R}^n; ...
1
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0answers
22 views

Derivative group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
1
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2answers
29 views

Does the signed measure based on a Jordan decomposition of a function with bounded variation depend on the decomposition?

Let $g_1, g_2, h_1, h_2 : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous. Define $$ \begin{align} f_1 & := g_1 - h_1 \\ f_2 & := g_2 - h_2 \end{align} $$ and suppose ...
2
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1answer
18 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta( \frac{x}{h}).$$ This ...
2
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0answers
22 views

convergence of systems of the form $V(t+T) = e^AV(t)$

Let assume I have a generic positive function $V:[0,\infty)\rightarrow \mathbb{R}^n$ which satisfy $$V(t+T) = e^{A} V(t),\qquad t\in[0,\infty)$$ where $A$ is a non diagonalizable real valued $n\times ...
1
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0answers
14 views

Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
1
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2answers
60 views

Are continuous functions with compact support bounded?

While studying measure theory I came across the following fact: $\mathcal{K}(X) \subset C_b(X)$ (meaning the continuous functions with compact support are a subset of the bounded continuous ...
4
votes
2answers
60 views

How do i find $\tan(\theta)$ such that : $\frac{16}{\sin^6(\theta)} + \frac{81}{\cos^6(\theta)}=625$??

How do i find $\tan(\theta)$ such that :$$\frac{16}{\sin^6(\theta)} + \frac{81}{\cos^6(\theta)}=625$$? Note : i used some trigono-form but sorry i didn't succed . Thank you for any help.
2
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2answers
38 views

Relationship between completeness and well ordering (meta).

Here is the definition for completeness of the reals (there are many equivalent formulations but I am interested in this one); Completeness: Every non-empty subset of the reals which is bounded above ...
3
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0answers
34 views

$L_{\infty}$ norm

For Lebesgue p-integrable functions, what would be the formula for $$\left(\int_0^1 \sum_{i=1}^n | f_i(x)|^p dx\right)^{\frac{1}{p}} $$ as $p\to +\infty$? Would it be $$\max_i \sup_{[0,1]} ...
1
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0answers
12 views

Question about sub differentiability of convex function

I am reading this book to study sub differentiability. On page 20 it says, as $V$ is a normed linear space for simplification, "a continuous affine function $l(v)$: $V\to \mathbb R$ everywhere less ...
2
votes
1answer
43 views

An open interval as a union of closed intervals

For $a<b, a,b\in\Bbb R$ $$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta] $$ Clearly the RHS is an (uncountable) infinite sum of closed intervals. I ...
3
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1answer
34 views

Existence of differentiable functions on $\mathbb R$ whose derivative is constant on the complement of uncountable set but not everywhere

Let $ A $ be a countable subset of the set of real numbers and $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f'$ is constant on $\mathbb R \setminus A$ , then I know that $f'$ is ...
16
votes
1answer
253 views

Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$

Are we aware of an elementary way of proving that? $$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$$ Of course, with the help of Mathematica it can be ...
6
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3answers
66 views

Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges

Let $x_n, y_n > 0$. I need an example of convergent sequences $x_n,y_n$ such that $x_n^{y_n}$ diverges. Could you help me?
4
votes
1answer
43 views

$f_n$ converges uniformly to f but $f^2_n$ fails to converge uniformly to $f^2$

Consider functional sequence $f_{n}$ which is differentiable on $\left(a,b\right)$. Find an example of $f_n$ converging uniformly to f on $\left(a,b\right)$ such that $f^2_n$ fail to converge ...
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3answers
38 views

Problem about $\sum_{n=1}^{\infty}n\exp\left(-x\sqrt{n}\right)$

Consider $$\sum_{n=1}^{\infty}n\exp\left(-x\sqrt{n}\right)$$ A. sum of the series is bounded on its set of convergence B. sum of the series is continuous on its set of convergence The correct ...
1
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1answer
19 views

Uniform convergence on singleton

First, recall the definition of uniform convergence: Consider functions $f_{n}:A\rightarrow\mathbb{R}$. The sequence of functions $f_{n}$ converges uniformly on set A to limit function f if ...
1
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2answers
34 views

What is the relationship between $L_{P}(0,1)$ and $L_{P}[0,1]$?

$L_{P}[0,1]$ be the set of measurable functions $f : [0,1]\rightarrow R$ such that $\int |f(x)|^{p} dx<\infty$. What is the relationship between $L_{P}(0,1)$ and $L_{P}[0,1]$?
1
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0answers
31 views

Combination estimation

I am looking at a proof where part of it derives an estimation for the number of combinations and I cant understand how the following step is derived: $\varepsilon^{-\varepsilon L}\underset{j \leq ...
2
votes
5answers
75 views

Question about using arbitrary $\epsilon$ in real analysis proofs

I've notice that in a lot of the proofs that are assigned in an undergraduate analysis course, we are often trying to show that some quantity is bounded by an arbitrary epsilon. For example, if I ...
4
votes
2answers
61 views

$\sum_{n=1}^{\infty} \frac{1}{n+1!} \prod_{k=1}^{n} f(k)$ Prove the divergence of a series [on hold]

How can I prove the divergence of the series $$\sum_{n=1}^{\infty} \left(\frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)\right) $$ if $f:\mathbb{N} \rightarrow \mathbb{N}$ is injective? $ $
2
votes
2answers
27 views

Compact sets are bounded: shape of the cover matters?

To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we ...