Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

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1answer
22 views

How to prove $\frac{7n^2 +5n}{3n^2-7}$ is convergent?

Prove the following converges: $$\displaystyle \lim_{ n \rightarrow \infty}\frac{7n^2 +5n}{3n^2-7} = \frac{7}{3}$$ Proposed Solution: Given $\varepsilon$ > 0, find a M $\in$ $\mathbb{N}$ s.t. $\...
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0answers
28 views

How to show this inequality $|x_k|\leq(\sum^{\infty}_{j=1}|x_j|^{p})^{1/p}$?

Is this inequality always true? How to prove this inequality $$|x_k|\leq(\sum^{\infty}_{j=1}|x_j|^{p})^{1/p}$$ Original post from the top answer The $ l^{\infty} $-norm is equal to the limit of the $ ...
6
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2answers
34 views

If a Taylor series agrees with its function in $U$, that function is analytic in $U$

Assume that $f:\mathbb{R}\rightarrow\mathbb{R}$ coincides with its Taylor series centered at a point $p\in U$ on an open set $U$, i.e. $$\exists p\in U, \forall x\in U, f(x)=\sum_{n=0}^\infty\frac{f^{(...
1
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1answer
28 views

Prove $\sup_{1 \leq n < \infty} \|\sum_{i=1}^n \alpha_i x_i\| \leq \lim_{n \rightarrow \infty} \sup_{1 \leq k \leq n} \|\sum_{i=1}^k \alpha_i x_i\|$

From book bases in Banach spaces I by ivan singer Proposition Let $\{x_n\}$ be a sequence in a Banach space $E,$ such that $x_n \neq 0 (n = 1,2,...),$ and let $Y$ be the Banach space of sequences of ...
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1answer
33 views

About unique zero of a polynomial of degree $4$: where is this theorem coming from?

Looking at the first comment of DavidP in the question How to understand for which $a, b\in\mathbb{R}$ the equation $a-x+\frac{b}{x^3}=0$ has a unique zero? I am wondering what you can tell that the ...
7
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1answer
68 views

Proving that $\{a\in\mathbb{Q}|a>0$ and $a^2<2\}$ has no least upper bound in $\mathbb{Q}$

I am going through a proof found in: http://www.math.columbia.edu/~harris/2000/2016Dedcuts.pdf In it he finds a smaller upper bound to the supposed least upper bound: proof However in Step 1, I don't ...
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0answers
19 views

Show that improper integral converges using taylor series

Show that the improper integral \begin{align*} \int_{0}^{x} \frac{\sin\left(t\right) }{t}\mathrm{~d}t .\end{align*} converges. My attempt: Determining the Taylor series of $\sin\left(t\right) $ ...
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2answers
25 views

Let $A$ be a closed set where every compact subset of $A$ is countable. Then prove that $A$ is also countable.

Let $A$ be a closed set where every compact subset of $A$ is countable. Then prove that $A$ is also countable. I've bee at this problem since this morning but I have literally no progress. First I ...
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0answers
10 views

Finding sets that are $\mu^*$ measurable

Let $E=\{\emptyset, (-3, 2], \mathbb{R}\setminus (-3, 2], \mathbb{R}\}$ and $\rho:E\to [0, \infty]$ such that $\rho(\emptyset)=0$, $\rho((-3, 2])=4$, $\rho(\mathbb{R}\setminus (-3, 2])=\rho(\mathbb{R})...
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1answer
34 views

Existence of $x_0$ in the exercise of Analysis

Let $f,g,h$ three real functions such that $$\lim_{x \to \infty}f(x)=\lim_{x \to \infty}g(x)=\lim_{x \to \infty}h(x)=0.$$ Furthermore, suppose also that $$\lim_{x \to \infty}\frac{f(x)}{h(x)}=0.$$ My ...
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0answers
58 views

Showing that $\{x\in\mathbb{Q}:x≤0 \ or \ x^2≤2\}$ does not have a least upper bound in $\mathbb{Q}$

I want to prove that $C=\{x\in\mathbb{Q}:x≤0 \ or \ x^2≤2\}$ do not have a least upper bound in $\mathbb{Q}$ The definition I'm working with is that a least upper bound, $b$ has to be an upper bound,...
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0answers
35 views

Equivalence of unit balls given the equivalence of norms

Let $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ be two different norms with corresponding unit balls $K_1$ and $K_2$, respectively. If $\Vert x \Vert_1 ≤ \Vert x \Vert_2$ for all $x ∈ R^d$, what ...
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0answers
38 views

What additional conditions are required for convergence in operator norm to imply convergence in trace norm

Let $\mathcal{H},\mathcal{K} $ be a separable Hilbert spaces, and $T_n: \mathcal{H}\mapsto \mathcal{K}$ a sequence of compact operators satisfying $\|T_n - T\|_{op} \to 0$, with $\|\cdot\|_{op}$ ...
2
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1answer
63 views

Are there any ordered fields containing $\mathbb{Q}$ as a non-dense subfield?

Does there exist a totally ordered field for which $\mathbb{Q}$ is not dense? The answer is yes, but I can't understand why. Can you provide an example?
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0answers
56 views

Alternate proof of the Intermediate Value Theorem

Let $f:[a,b]\to\mathbb{R}$ be continuous with $f(a)<0<f(b)$. Let $p=\inf\{t\geq a,f(t)\geq 0\}$ be the first time $f$ reaches $[0,\infty)$. Prove that $p\in(a,b)$ and that $f(p)=0$. It makes ...
2
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0answers
31 views

Prove that uniform convergence almost everywhere implies convergence almost uniformly?

How can I prove the following result? Uniform convergence almost everywhere implies convergence almost uniformly. I know Egorov's theorem which says pointwise convergence almost everywhere implies ...
2
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1answer
29 views

The total variation of a continuous $f$ satisfies: $\lim_{\epsilon \to \infty} T_\epsilon ^1(f) = T_0^1(f)$.

This is essentially a supremum and limits question. I've encountered the following claim: If $f$ is continuous on $[0,1]$, then $f$ satisfies: $\lim_{\epsilon \to 0^+} T_\epsilon ^1(f) = T_0^1(f)$. ...
2
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0answers
38 views

About integration of $\frac{1}{(1+a^2 x^2)(1+x^2)}$ in differentiation under integral sign

Consider the function $F(a)=\int_0^{+\infty} \frac{\arctan(ax)}{x(1+x^2)} dx$. I assumed $a \ge 0$ because $F$ is odd, I applied the theorem of differentiation under the integral sign and arrive to $F'...
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0answers
26 views

A Grönwall-type inequality for $u(t)\le\alpha(t)+\int_0^t\max(u(s),\beta(s))\:{\rm d}s$?

Quick question: Are we able to show a Gronwall-type inequality assuming that $$u(t)\le\alpha(t)+\int_0^t\max(u(s),\beta(s))\:{\rm d}s,$$ where $\alpha$ is nondecreasing (or constant) and $\beta$ is ...
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0answers
43 views

Countable sum of Baire of class 2 functions on $\Bbb R$

For every $k\in\Bbb Z$, let $f_k$ be a Baire class function on $\Bbb R.$ Assume $\sum_{k\in\Bbb Z} f_k$ is convergent. Define $f:=\sum_{k\in\Bbb Z} f_k$ so $f$ is a function. Moreover, $f$ is a Baire ...
2
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0answers
67 views

Proving $\int_{0}^{1} \frac{K(x)K\left ( \sqrt{2} \sqrt{(1-x^2)/(2-x^2)} \right ) }{2-x^2}\text{d}x =\frac{\pi^3}{8\sqrt{2} } {}_6F_5(...)$

I encountered an integral identity: $$\int_{0}^{1} \frac{K(x)K\left ( \sqrt{2} \sqrt{\frac{1-x^2}{2-x^2} } \right ) }{2-x^2}\,dx =\frac{\pi^3}{8\sqrt{2}} {}_6F_5\left ( \frac{1}{4},\frac{1}{4},\...
1
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0answers
7 views

Are jumps of $C^2$ cadlag functions finite variation? Why are the discontinuous parts of Ito formula of finite variation?

Below is an excerpt from the proof of Ito's formula in the following post :https://almostsuremath.com/2010/01/25/the-generalized-ito-formula/#scn_genito_eq3 We assume $f$ to be a real $C^2$ function ...
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1answer
47 views

Is $f_k(x) := k \sin \left(\frac1x\right) x^{k-1} - \cos\left(\frac1x\right) x^{k-2}$ continuous at zero?

I'm trying to improve my proof writing due to exams. Is my reasoning even correct in the first place and understandable? If not what could be improved? Let $f_k$ be a set of functions from the real ...
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0answers
16 views

Derivate of the the negative log likelihood with composition

We want to solve the classification task, i.e., learn the parameters $\theta = (\mathbf{W}, \mathbf{b}) \in \mathbb{R}^{P\times K}\times \mathbb{R}^{K}$ of the function $f_\theta: \mathbb{R}^P \to [0, ...
3
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1answer
36 views

A question about the "dominated" part of the DCT

Let $f_n:\mathbb{R}\to [0, \infty)$ be a sequence of measurable functions that converges to some function $f$. Suppose that I show that for "$n$ big enough" we have $|f_n(x)|\le g(x)$ for ...
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0answers
28 views

Grönwall's lemma for $t < t_0$

Let $T \in \mathbb{R}^+ \cup \{ \infty\}, \,t_0\in [0,T), \,a,b\in L^{\infty}(t_0, T)$ and $\lambda \in L_1(t_0, T), \lambda(t)\geq 0$ for almost all $t\in (t_0, T).$ Then, it follows from $$a(t) \leq ...
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0answers
54 views

Find the minimum $k ∈ \mathbb N$ such that $\sum_{n=p}^{\infty}{\frac{1}{\log(n)\log^{(2)}(n)\dots\log^{(k)}(n)}} $ converges

I will need some help with the following question: Find the minimum $k ∈ \mathbb N$ such that $\sum_{n=p}^{\infty}{\frac{1}{\log(n)\log^{(2)}(n)\dots\log^{(k)}(n)}} $ converges where $\log^{(k)} = \...
0
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0answers
29 views

Arc length of a sine wave kinda

I find myself trying to work out a formula to help in speeding up a task i do on a daily basis, but it seems to be beyond my education. I'm trying to work out the length of a sort of sine wave that ...
1
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1answer
41 views

Is $\lfloor x \rfloor + \sqrt{ x - \lfloor x \rfloor} $ continuous on $\mathbb{R}$?

Honestly I don't know how to proceed. The solution in my textbook proves that the function is continuous at $0$ (which is easy to show) and then concludes that it must be continuous everywhere. Why? ...
0
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0answers
17 views

Does the Riemann-Liouville integral always exist for any Lebesgue integrable function $f$?

Let $ J= [a, b],\, (-\infty<a<b<+\infty)$ be a finite interval on the real axis $\mathbb{R}$. In many monographes, i find that the Riemann-Liouville fractional integral of the function $f \in ...
-2
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0answers
22 views

How to find that a convergent sum is equal to? [closed]

Suppose that $\sum a_k$ al is absolutely convergent, show what: $\sum_{k=1}^{\infty} a_k = \sum_{k=1}^{\infty}a_{2k-1} + \sum_{k=1}^{\infty}a_{2k} $ I have to resolve this but honestly I don't know ...
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0answers
58 views

If a monotone function $f:[a,b]\to \mathbb R$ is differentiable on a closed interval $[a,b]$, is $f'$ bounded?

I first tried to find a counterexample $\sqrt{x}$ on $[0,1]$, but $\sqrt{x}$ is not differentiable at $0$. I also kwon the function $$f(x) \;=\; \begin{cases}x^2 \sin (1/x^2) & \text{if }x\ne 0, \\...
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0answers
5 views

Clarification on measurability and a particular set cover

Preamble: Suppose that $m^*$ is the Radon outer measure on $\mathbb{R}^n$ and $S \subseteq \mathbb{R}^n$. Define $r_i^*(S) = (m^*|B_i(0))(S) = m^*(S\cap B_i(0)), S \subseteq \mathbb{R}^n$ the ...
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0answers
24 views

Bounded bivariation & bounded from zero doesn't imply that reciprocal is of bounded bivariation

Let $I, J \subset \mathbb{R}$ be compact intervals. We call $f\colon I \times J \to \mathbb{R}$ is of bounded bivariation if for every partition $\{x_0, \ldots, x_m\}$ of $I$ and $\{y_0, \ldots, y_n\}$...
3
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1answer
74 views

How to understand for which $a, b\in\mathbb{R}$ the equation $a-x+\frac{b}{x^3}=0$ has a unique zero?

Let $a, b\in\mathbb{R}$, $x\in\mathbb{R}^*$ and consider te he equation $$a-x+\frac{b}{x^3}=0.$$ My question is: there is a way to understand for which values of $a, b$ has a unique zero? I tried by ...
2
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1answer
89 views

Is connectedness subspace invariant?

If $X$ is a topological space and $Y$ is a subspace of $X$, then is it true that any subset $Z$ of $Y$ which is connected in the topology of $X$, is also connected in the subspace topology of $Y$? I ...
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0answers
46 views

Probe that $⌊x-y⌋ ≤ ⌊x⌋ - ⌊y⌋ ≤ ⌊x-y⌋ + 1$ , floor function ⌊..⌋

Can someone give me an idea to demonstrate that: $$⌊x-y⌋ ≤ ⌊x⌋ - ⌊y⌋ ≤ ⌊x-y⌋ + 1$$ where $⌊x⌋$ is the floor function of $x$ Thanks in advance.
0
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1answer
43 views

General formula for the integral of $f(s)^k$ with respect to $s$

Consider the definite integral \begin{equation} \mathcal{I}=\int_{-\infty}^t f(s)^k\mathop{ds}. \end{equation} In terms of $F(t):=\int_{0}^t f(s)\mathop{}ds$, is there a simple expression for $\...
0
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0answers
55 views

A "twisted" limit of the function $F(x,y)$

Let $F:\mathbb R^2 \to \mathbb R$ be a function (not necessarily continuous). Suppose for any $y>0$, we have $\lim_{x\to +\infty}F(x,y)=0$. Does there necessarily exist a positive-valued function $...
3
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1answer
70 views

$\max_{k=1}^{n}|x_k| \xrightarrow[\text{}]{\text{$n \rightarrow \infty$}} \sup_{k=1}^{\infty}|x_k|$

I want to prove the following: Let $(x_n)$ be a bounded sequence, then $\max_{k=1}^{n}|x_k| \xrightarrow[\text{}]{\text{$n \rightarrow \infty$}} \sup_{k=1}^{\infty}|x_k|$ . My Calculations: Let $c:=\...
1
vote
1answer
57 views

Can 2 different ODE's have the same set of solutions?

If I have two differents linear ODE's: \begin{equation} x'' + p(t)x' + q(t)x = f(t) . \quad p,q \in C(I,\infty). \\ x'' + j(t)x' + g(t)x = h(t). \quad j,g \in C(I,\infty). \end{equation} Coul they ...
-3
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1answer
52 views

Show, without calculating the integral, that $\int^\infty_2 \frac{1}{\sqrt[3]{x^3-1}}$ is convergent [closed]

I'm a bit stuck on this problem at the moment. How do I approach something like this? "Show, using appropriate estimations, and without calculating the integral, that $\int^\infty_2 \frac{1}{\...
1
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1answer
38 views

Points for which $f_n(x) = \sum_{k=1}^n |\sin(\pi k! x)|^{1/n}$ is bounded

For a sequence of positive continuous functions $(f_n)$, let's define $E$ as $\{x: (f_n(x))$ is bounded$ \}$. I need to find $E$ for the sequence of functions $f_n(x) = \sum_{k=1}^n |\sin(\pi k! x)|^{...
0
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2answers
42 views

Justifying formaly that $ \inf\{c\ge 0: \forall x\in V,\|Tx\|\le c\|x\| ,\forall x \in V\} =\sup\{\|Tx\|/\|x\|\: \forall x\in V,x \neq 0 \}$

$$\|T\|_1=\inf\{c\ge 0: \|Tx\|\le c\|x\| ,\forall x \in V\}=\inf\ S_1$$ $$\|T\|_2=\sup\{\|Tx\|/\|x\|\: \forall x\in V,x \neq 0 \}=\sup S_2$$ I am trying to formalize the equality between these two ...
2
votes
1answer
60 views

How can i simplify the following formula: $\sum\limits_{i,j=1}^{n}(t_{j}\land t_{i})$?

Consider the following time discretization $t_{0}=0< t_{1} < ... < t_{n} = T$ of $[0,T]$ where the time increments are equal in magnitude, i.e. $t_{j}-t_{j-1}=\delta$. How can i simplify the ...
0
votes
1answer
65 views

A function $f(x)$ is uniformly continuous on $(a,b)$ $\iff$ it can be defined on $\{a, b\}$ such that $f$ is continuous on $[a,b]$.

A Theorem regarding Uniform Continuity states : A function $f(x)$ is uniformly continuous on $(a,b)$ $\iff$ It can be defined on the endpoints $a$, $b$ such that $f$ is continuous on the $[a,b]$. ...
0
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0answers
17 views

Calculating the radius of convergence for a power series with coefficients fulfilling $c_{j+2} = \frac{j^2+j-L}{(j+1)(j+2)}c_j$, $L \in \mathbb{R}$

I am considering the power series $y(x) := \sum_{j=0}^{\infty} c_jx^j$. The coefficients fulfill \begin{align} c_0 = 0, \quad c_1 = 1 \quad \text{ and } \quad c_{j+2} = \frac{j^2+j-L}{(j+1)(j+2)}c_j, ...
1
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0answers
44 views

Proving a function is in $L^{\infty}$

Let $f:\Bbb{R^2}\to\Bbb{R}$ be a lebesgue measurable function, $f\in L^{\infty}$. For every $\theta\in [0,2\pi)$, define $g_\theta(x)=f(x\cos(\theta),x\sin(\theta))$. Prove that for almost every $\...
4
votes
2answers
61 views

Prove that the composition of continuous functions is continuous using the $\epsilon-\delta$ definition of continuity.

Definition of Continuity: A function $f:A \rightarrow \mathbb{R}$ is continuous at a point $c \in A$ if, for all $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x - c| < \...
2
votes
1answer
42 views

Inverse of $\Phi(f) =\operatorname{sgn}(f)\lvert f\rvert^{1/p}$ is continuous from $L^p(E)$ to $L^1(E)$

I am working through Real Analysis, fourth edition, by Royden and Fitzpatrick. I ask your help with the last sentence, in emphasized font below, of Problem 48 of Chapter 7. I have included two lead-up ...

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