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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4
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1answer
62 views

$f(x) =\lim_{n \to \infty} \frac{(1+ \sin \frac{\pi}x)^n - 1} { (1+ \sin \frac{\pi}x)^n +1}$, $x \in (0,1]$. To show that $f$ is integrable on $[0,1]$

A function defined on $[0,1]$ by $f(0) = 0$ and $f(x) = \lim_{n \to \infty} \frac{(1+ \sin \frac{\pi}x)^n - 1} { (1+ \sin \frac{\pi}x)^n +1}$, $x \in (0,1]$. To show that $f$ is integrable on ...
1
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3answers
43 views

Proof that boundedness of continuous Real Valued functions implies Compactness

I'm looking to prove the following : Let $(X,d)$ be a Metric Space If every continuous real-valued function on $X$ is bounded then $X$ is Compact I saw a proof earlier today If instead $X$ is ...
0
votes
2answers
65 views

Is my proof for this limit correct?

I want to prove that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to 2. Let $a_0$ = $\sqrt{2}$ $a_n$= $\sqrt{2+a_{n-1}}$. Then, proving that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to ...
0
votes
1answer
69 views

Prove that $[f(x+1)-f(x)] = 0$

I know this is a very elementary level question. But, I still need t0 understand this in term of mean value theorem. So here it goes: Suppose $f$ is differentiable on $(0,\infty)$ and lim$_{x \to ...
3
votes
1answer
36 views

Variation processes and strong solutions of stochastic differential equations

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$ $\tau$ be a $\mathbb{F}$-stopping time An $\mathbb{F}$-adapted, ...
0
votes
4answers
66 views

Why is this function a bijection?

Consider the function below $$f:\mathbb{R^+} \to \mathbb{R^+}$$ given by $$f(x) = \sqrt{x}$$. Now it makes sense that the function is injective because $f(x) = f(y) \implies \sqrt{x} = \sqrt{y} ...
1
vote
1answer
54 views

Show Lie bracket left invariant

I want to prove from the definition that the Lie bracket $[X,Y]$ of two left-invariant vector fields $X,Y: G \rightarrow TG$ where $G$ is a Lie group is again left-invariant. Left-invariance ...
1
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0answers
17 views

True/False? If $a ∈ iso(S)$ , then, $a_i ∈ iso(π_i(S))$ for all $i ∈ \mathbb N_n,$ where $π_i$ denotes the natural projection of $P$ onto $X_i$

Suppose $n ∈ \mathbb N$ and, for each $i ∈ \mathbb N_n, (X_i, τ_i)$ is a metric space. Suppose $d$ is a conserving metric on $P = \prod_{i=1} ^n X_i .$ Suppose $S ⊆ P$ and $a ∈ S.$ Is it true that If ...
2
votes
3answers
96 views

Prove that $f'(c ) = \lambda f(c )$

Suppose $f$ is continuous on $[a,b]$ and $f$ is differentiable on $(a,b)$ with $f(a) = f(b) = 0$. Prove that for every real $\lambda, \exists c \in (a,b)$ s.t. $f'(c ) = \lambda f(c )$. Hint: Apply ...
3
votes
1answer
45 views

Surjectivity of $\mathcal{id}_{\mathbb{R}^n}+g$ when $g$ is a contraction?

Assume $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is a contraction and consider $h=\mathcal{id}_{\mathbb{R}^n}+g$. The map $h$ is injective. Is it always surjective? My question has the following ...
4
votes
3answers
92 views

Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$?

Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$, or to be accurate: why is $\int_{\pi\over 2}^{\pi}{1\over 1-\sin x}dx=\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$? ...
1
vote
0answers
79 views

Distance between sets

Let $K \subset K_1 \subset U \subset \Bbb R^2$, such that $K$ and $K_1$ are compact sets, with $K \subset \mathring {K_1}$, and $U = \mathring U \subsetneq \Bbb R ^2$. If $w \in \partial K_1$ such ...
1
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0answers
18 views

Prove that $\frac{\langle f^2,g\rangle_{L^2}}{\left\|f\right\|_{L^2}^2}\ge-\left\|g\right\|_{L^\infty}$ for $f\in L^2$ and $g\in L^\infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $g\in L^\infty(\Omega)$. How can we show, that $$\frac{\langle ...
0
votes
2answers
34 views

C1 function with strictly positive derivative at its root(s); can I prove that $x<y,\;f(x) > 0 \Rightarrow f(y) > 0$?

My question is somehow related to Positive derivative at root of $f$. but yet slightly different. Let $f$ be a C1 function in $(a,b) \subseteq \mathbb{R}$, not necessarily monotonic but with the ...
2
votes
0answers
42 views

Orthonormal Basis of $L^2$

Theorem: ' ' The Orthonormal family $e_n(x)=e^{2\pi i n x},\ n\in\mathbb{N}$ is a basis for $\mathcal{L}^2([0,1])$.`` In this case, $\{e_n(x)\}_{n\in\mathbb{N}}$ being a basis would mean that any ...
2
votes
1answer
102 views

Prove that $x+g$ is homeomorphism

Problem: Assume we have $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ of $C^1$ class with derivative bounded uniformly by some constant $M<1$. Consider ...
0
votes
1answer
15 views

Definition of “Contractive Invariant Plane”

Can someone please explain the definition of a contractive invarient Plane found in: the paper It is nearly at the very beginning of the Introduction. By contractive do they mean a contractive map? ...
1
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0answers
15 views

Proposed proof for quasi-metric result

A quasi-metric on a set $X$ is mapping $\rho: X \times X \rightarrow [0, \infty)$ satisfying the following conditions: $\rho(x,y) \geq 0~~\text{and}~~\rho(x,x) = 0;$ $\rho(x,z) \leq \rho(x,y) + ...
1
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0answers
28 views

Series Convergence in Banach Space

Let $(e_j)_1^\infty$ be an orthonormal set in $l^2$ Consider $$s_n =\sum_{j=1}^n t_je_j$$ Show that $s_n$ converges in $l^2 \iff t = (t_j)_{j=1}^\infty \in l^2$ Thoughts so far : If we consider ...
2
votes
1answer
26 views

What is the convex hull of $ \{t \to e^{-\lambda t} : \lambda >0\}? $

What is the convex hull of $$ \{t \to e^{-\lambda t} : \lambda >0\}? $$ (Interpreted as the set of all functions on the above form.) Reference or argument is great.
1
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1answer
31 views

Proof Strategies for Convergent Sequences

I am struggling to understand how to choose epsilons during proofs for convergent sequences. It seems that many proofs just state the epsilon to choose without any motivation? How should I go about in ...
2
votes
2answers
134 views

Evaluate the improper integral $\int_{0}^{\infty}{f(x)-f(2x)\over x}dx$, where $\lim_{x \to \infty} f(x) = L$ [duplicate]

Find $$\int_{0}^{\infty}{f(x)-f(2x)\over x}\, \mathrm{d}x$$ if $f\in C([0,\infty])$ and $\lim\limits_{x\to \infty}{f(x)=L}$. I tried denoting $\displaystyle \int{f(x)\over x}dx=F(x)$, but I don't ...
2
votes
2answers
44 views

Converge series such that permuting the termes will change the limit.

I know that for a series that converge, if we permute the element of the sum, the series doesn't necessarily converge. For exemple $$\sum_{n=1}^\infty \frac{(-1)^n}{n}$$ converge but if we first sum ...
0
votes
1answer
35 views

limit of functions and the integration

Let $f_n:\mathbb{R}\rightarrow\mathbb{R}$ be a sequence of non-negative Lebesgue measurable functions and suppose $\lim_{n\rightarrow \infty}\int_\mathbb{R}f_n=0.$ Then, must it be that ...
5
votes
1answer
118 views

Definite integral with logarithm and arctangent inside of arctangent

How to prove $$\int_0^1 \left[ \frac{2}{\pi }\arctan \left(\frac 2 \pi \arctan \frac{1}{x} + \frac{1}{\pi }\ln \frac{1 + x}{1 - x}\right) - \frac{1}{2} \right]\frac{\mathrm{d}x} x = \frac{1}{2} \ln ...
0
votes
2answers
31 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
votes
1answer
67 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 ...
0
votes
0answers
49 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. ...
1
vote
3answers
32 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
votes
2answers
19 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
votes
1answer
34 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 ...
3
votes
4answers
57 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 ...
1
vote
1answer
36 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 ...
0
votes
1answer
31 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 ...
-1
votes
0answers
55 views

Limit of a recursive sequence containing log [closed]

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 ...
6
votes
1answer
106 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 ...
1
vote
1answer
30 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
19 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
vote
3answers
46 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 ...
1
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0answers
35 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
votes
0answers
50 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: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
3
votes
1answer
121 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
28 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
votes
1answer
63 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
vote
1answer
46 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
vote
2answers
43 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 ...
-3
votes
0answers
40 views

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

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
votes
3answers
74 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
votes
0answers
62 views

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

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
votes
1answer
54 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 ...