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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1answer
9 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that ...
-1
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0answers
26 views

Show $g$ is measurable

Suppose $f$ is continuous in $\mathbb R$ such that $|\{x:f(x)=y\}|<\infty$ for any $y\in \mathbb R$. Show that $$ g(y)=|\{x:f(x)=y\}| $$ is a measurable function. Moreover, show this is not true if ...
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0answers
29 views

Two ODEs, why is one solution the solution of the other?

Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ Caratheodory, and we know that if it has a solution, it ...
6
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1answer
25 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
0
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1answer
16 views

$L^1 ([0,1])$, bouned linear functional, absolute continuous function

I am studying for an Analysis prelim and was wondering if someone could perhaps either validate or invalidate my proof for the following problem: "Let $L^1 ([0,1])$ be the space of Lebesgue ...
1
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0answers
26 views

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
0
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4answers
45 views

Must a continuous function on $\mathbb R$ with only rational values be constant? [duplicate]

As I'm preparing for my exam I have to solve the following question: Determine if the following is correct: Let $f$ be a continuous function is $\Bbb R$. If $f$ recieves only rational values, ...
0
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0answers
29 views

Induction on derivatives

I have troubles understanding this induction proof: Let $$g(x) = \vert x \vert^{2k+1}$$ Show by induction: $$\frac{\partial ^N g(x)}{\partial x_{i_1} \dots \partial x_{i_N})} = cx_1n \dots x_iN \vert ...
4
votes
1answer
32 views

Condition for Continuity (two variable)

I came across the following question while studying for quals. This one is from a previous qualifier. I have a few ideas (which I'll mention below), but am stuck on how to complete the problem. Any ...
2
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1answer
25 views

If a measure $\mu$ is less than a measure $\nu$ on a generating $\pi$-system, can we conclude that $\mu \leq \nu$?

Let $\mu$, $\nu$ be finite measures on the non-degenerate compact interval $[a, b] \subseteq \mathbb{R}$ provided with the Borel $\sigma$-algebra. It is well-known that if $\mu(B) = \nu(B)$ for every ...
1
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2answers
16 views

Conditional convergence and Riemann's series theorem

There are tests to determine whether an integral or sum is convergent. There are test to determine whether an integral or sum is absolutely convergent. An integral or series is said to be $\mathbf ...
2
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1answer
54 views

How to prove that $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $?

If $n \in \mathbb N$ and $n \geq 2$, then we have $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $. My try : Once if we can prove that for all $k \in ...
3
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2answers
51 views

Is this statement “a map $f$ is continuous if and only if for any open set $G$, ${f^{ - 1}}(G)$ is still open” true?

I am really puzzled by this statement and it has so many different versions in different places. Yesterday I did a homework to prove that a finite function $f$ is continuous if and only if ${f^{ - ...
3
votes
3answers
56 views

Why is $\max(x, x')$ equivalent to $\frac{1}{2}( x + x' + |x - x' |)$?

Why is it that $$\max(x, x') = \frac{1}{2}( x + x' + |x - x'|)$$ is true? Is it supposed to be obvious? Because it seems to come out of thin air for me. Anyway, I've verified this by plotting it in ...
1
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1answer
14 views

Open condition given by inequality on functions

Let's say we have two functions $f,g\in C^\infty(D)$, $D$ an open domain in $\mathbb{R}^2$. The condition $f(x,y)<g(x,y)$ is an open condition on $D$? With this I mean: do the points $(x,y)\in D$ ...
0
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1answer
12 views

Upper semi-continuity results

I have recently been introduced to the notion of upper semi-continuity on a metric space $X$. Please advise on the following queries: If $f:X \rightarrow \mathbb{R}$ is upper semi-continuous and ...
0
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0answers
17 views

Proving Squeeze Theorem using Order Limit

Would this be a valid way of proving the squeeze theorem using the Order Limit theorem? If $x_n \leq y_n \leq z_n$ for all $n \in \mathbb{N}$ and if $\lim x_n =\lim z_n = l$, then $\lim y_n = l$ as ...
1
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1answer
97 views

Is $(\frac 1{n^2 \sin n })$ convergent ? If so , what is the limit? [duplicate]

Is the sequence $\left(\dfrac 1{n^2 \sin n }\right)$ convergent ? If it does, with what limit?
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0answers
34 views

Principle of infinite descent

I'm trying to prove the following: "It is not possible to have a sequence of natural numbers which is in infinite descent. (Hint: assume for sake of contradiction that you can find a sequence of ...
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0answers
15 views

Periodicity of Newton's method approximations on a cubic polynomial

Bruckner & Bruckner, Elementary Real Analysis Let $f(x) = x^3 - 3x + 3$ Applying Newton's method to get $x_{n+1} = x_n -\frac{f(x)}{f'(x)} \ ,$ prove that for any positive integer $p$, there ...
1
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1answer
18 views

If $(B_t)_{t\ge 0}$ is a Brownian motion and $\tau$ is a stopping time, then the stopped process $(B_{\min(\tau,t)})_{t\ge 0}$ is integrable

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$. By definition $B_t$ is normally distributed with mean $0$ and variance $t$. Now, let ...
0
votes
1answer
23 views

Application of inverse function theorem?

I am not completely sure if this a direct consequence of the inverse function theorem. Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of ...
0
votes
4answers
46 views

Proof of Convergence of a Sequence

Show that the sequence $\frac{n^2+1}{n^2+n}$ converges and its limit is $1$. However, I am finding it difficult to prove according to the rules that a converging sequence must obey, that is, sequence ...
0
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2answers
36 views

$f: [0,1] \to \mathbb R$ is continuous and $\int_0^x f(t) dt = \int_x^1 f(t)dt$ for all $x \in [0,1]$, then $f(x) = 0$ for all $x \in [0,1]$.

A function $f: [0,1] \to \mathbb R$ is continuous on $[0,1]$ and $\int_0^x f(t) dt = \int_x^1 f(t)dt$ for all $x \in [0,1]$, then $f(x) = 0$ for all $x \in [0,1]$. My Try: Let us assume that $f(x) ...
2
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1answer
25 views

$ \lim_{\gamma_- ~\to ~-1} \Phi\left(\frac{-\gamma~ \Phi ^{-1}(\alpha) - \Phi ^{-1}(\beta)} {\sqrt{1- \gamma^2}} \right)$

I am struggling to calculate the following linmit $$ \lim_{\gamma_- ~\to ~-1} \Phi\left(\frac{-\gamma~ \Phi ^{-1}(\alpha) - \Phi ^{-1}(\beta)} {\sqrt{1- \gamma^2}} \right)$$ where $\alpha,\beta \in ...
0
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0answers
18 views

Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
12
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2answers
388 views

Are eigenvalues of the limit of a sequence of matrices limits of eigenvalue sequences?

Let $\{A_n\}\in \mathbb{R}^{m\times m}$ be a sequence of symmetric matrices such that $A_n\to A$ as $n\to \infty$, i.e. $\lim_{n\to \infty}a_{ij}(n)=a_{ij}\ \forall 1\le i,j\le m$ where ...
-3
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1answer
33 views

For what values of α the following serie converges?

For what values of α the following series converges? $\displaystyle\sum_{n=1}^{\infty} (\frac{1}{n}-\sin\frac{1}{n})^{\alpha}$ Help.. Thanks...
0
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0answers
11 views

Optional stopping/sampling for right-continuous supermartingales

Let $\mathbb{F}$ be a filtration $(X_t)_{t\ge 0}$ be a right-continuous $\mathbb{F}$-supermartingale $\sigma,\tau$ be bounded $\mathbb{F}$-stopping times with $\sigma\le \tau$ and ...
10
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2answers
156 views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\dfrac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...
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0answers
20 views

construction of a path of quadratic variation

Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) = \lim_{|\Pi| \to 0} \sum_{i=1}^{n}|x(t_{i}) - x(t_{i-1})|^{p}$$ where $\Pi = \{a= ...
0
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0answers
28 views

Help with Fourier transform of product

I was reading this article in wikipedia, and I supposed $f,g \in L^1(\mathbb{R^n})$ such that their product $f \cdot g$ are in $L^1(\mathbb{R^n})$ too. So let $h=f \cdot g$, and ...
1
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3answers
39 views

How to determine the closure of a subset and prove it is actually the closure?

I have this subset $E = \{r \in Q: r^2 \leq 2\}$ which is in $\mathbb{R}$ with the Euclidian metric. I was wondering how can I find the closure of this subset. Here is what I have: The limit points ...
-4
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3answers
76 views

Prove that $f(x) = x$ has a solution on [0,1] [duplicate]

Let $f:[0,1]\to [0,1]$ be a continuous function. Prove that $f(x) = x$ has a solution in $[0,1]$. So, here, $f(0) = 0$ and $f(1) = 1$ and $f'(x) = 1$. Now, can we conclude that there is at least one ...
4
votes
1answer
57 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
41 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
64 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
63 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
31 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
61 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
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1answer
40 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 ...
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0answers
16 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
39 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
87 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 $\int_{\pi\over 2}^{\pi}{1\over 1-\sin x}dx=\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$? ...
2
votes
0answers
72 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
vote
0answers
12 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
33 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
36 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 ...
1
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0answers
63 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 ...