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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8 views

Does this sequence converge? If yes, what is the limit?

Assume $\{k_n\}_{n\geq 0}$ a sequence of natural numbers such that $k_0=0$, $k_n\leq k_{n+1}\leq k_n+1$, and $\lim_{n\rightarrow\infty} \frac{k_n}{n}=\alpha\in(0,1)$. So $\{k_n\}$ is an ...
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0answers
18 views

Showing that this bounded linear map is invertible

Let $T$ be a bounded linear operator from the Hilbert space $H_1$ to a Hilbert space $H_2$. Suppose there exists $\delta>0$ such that $$ \langle T^{*}Tx,x\rangle\ge \delta \|x\|^2 $$ for all ...
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1answer
17 views

Unsolved regular value problem in $\mathbb{R}^n$

I want to show that if $F : \mathbb{R}^n \rightarrow \mathbb{R}^{n-k}$ is a $C^1$ function and $rank(DF) = n-k$ then $M:=F^{-1}(\{0\})$ defines a manifold. My idea: Without loss of generality I ...
1
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1answer
20 views

If $T^{*}$ is injective then $T$ is surjective?

If $T$ is a bounded linear map from the Hilbert space $H_1$ to the Hilbert space $H_2$, and $T^{*}$ is injective, then I know that $H_2$ is the closure of the range of $T$. But can I conclude that $T$ ...
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1answer
14 views

Convergence in $L_{\infty}$

Let $(X,\Sigma,\mu)$ be a measure space and $E_n\in\Sigma$ such that $\mu(E_n)>0$, and we define $f:=a_n\chi_{E_n}$, where $a_n>0$ for each $n\in\mathbb{N}$. I want to prove that if $f_n\to 0$ ...
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0answers
13 views

compactness of $K$ and $(K,d)$

I am trying to prove that if $(X,d)$ is a metric space and $K \subset X$ then $K$ is compact iff it is sequentially compact. The forward implication has been proved I am looking at the proof for the ...
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10 views

Properties of the space of T-􀀀invariant probability measures over a compact topological space.

Let $T: X \to X$ be a continuos map defined on the compact space $X$. Denote by $M_T$ the space of $T$-invariant probability measures defined over the Sigma Algebra of Borel sets. Prove the ...
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2answers
17 views

Convergence Sequence on compact set

Let $A\subset\mathbb{R}^n$ be a compact and $x\in A$. Suppose that $x_n$,$n\in N$ is a sequence in $A$ with the property that every convergent subsequence of $x_n$ converges to $x$. Prove that the ...
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0answers
12 views

Do smooth solutions of $u_{t}(x, t) = \Delta u_{t} + u$ satisfy $\sup_{0 \leq t \leq T}\|u(\cdot, t)\|_{L^{2}_{x}} = \|u(x, 0)\|_{L^{2}}$?

Let $u(x, t) : \mathbb{R}^{d} \times [0, T) \rightarrow \mathbb{R}$ be a smooth solution to $$u_{t}(x, t) = \Delta u_{t} + u$$ with $u(x, 0) = u_{0}(x) \in L^{2}(\mathbb{R}^{d})$. Furthermore, suppose ...
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1answer
26 views

Is the following a valid characterisation of complete metric spaces?

A metric space $(X, d)$ is called complete if and only if every Cauchy sequence converges. Now does the following hold: A metric space is complete if and only if every sequence $(x_i)_{i\in\mathbb ...
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0answers
23 views

Taylor expansion of the solutions of the equation $1-4 \cos(\frac{1}{x})+8x \sin(\frac{1}{x})=0$

In following article, I give an example of a function whose derivative at 0 is equal to 1 but which is not increasing near 0. The function is: $$\begin{array}{l|rcl} f : & \mathbb{R} & ...
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6answers
67 views

Prove that if $\lim_{n\to\infty}\frac{x_{n+1}}{x_n} = L < 1$ then $\lim_{n\to\infty} x_n = 0$

Let $(x_n)$ be a sequence with $x_n > 0$ for all $n \in \mathbb{N}$. I would like a hint on how to prove that if $\lim_{n\to\infty}\frac{x_{n+1}}{x_n} = L < 1$ then $\lim_{n\to\infty} x_n = 0$. ...
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1answer
19 views

Characterization of invertibility of bounded linear operator between Hilbert spaces

Let $T$ be a bounded linear operator from the Hilbert space $H_1$ to a Hilbert space $H_2$. I've shown that the existence of a $\delta>0$ such that $$ \langle T^{*}Tx,x\rangle\ge \delta \|x\|^2 ...
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3answers
61 views

Analyzing if function is “onto”

I have some function $g$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $g: [a,b] \to [a,b]$ is onto. How can I find out if this is true or not? P.S. I am not saying all $g$ ...
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3answers
81 views

How to evaluate $\lim _{x\to 0}\frac{1}{x^5}\cdot \int _0^x\:f\left(t\right)dt$ with MVT

We have to evaluate $\lim _{x\to 0 }\frac{1}{x^5}\cdot \int _0^x\:f\left(t\right)dt$ where $f\left(x\right)=x-\frac{x^2}{2}+\frac{x^3}{3}-\log\left(1+x\right)$ , $\:x\in \left(-1,\infty \right)$ I ...
0
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1answer
11 views

Convexity/concavity of a strictly increasing and continuous function

Consider a continuous, strictly increasing function $f:\mathbb{R}_{+}\to \mathbb{R}_{+}$ with $f(0)=0$, and $x>f(x)$ for all $x>0$. Is this enough to conclude anything about ...
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2answers
33 views

Analyzing derivative of function.

I have some function $g: [a,b] \to [a,b]$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $\forall x \in [a,b]: |g'(x)| \lt 1$. How can I find out if this is true or not? P.S. I ...
1
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1answer
24 views

Formula for the Beta function for natural m, n

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function $B(x, y)$, it's symmetry $B(x,y) = B(y,x)$ aswell as the fact that $(x + y)B(x + 1, y) = xB(x, y) ...
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2answers
20 views

Monotonous everywhere function

$f: \mathbb R \to \mathbb R,\forall x \in \mathbb R $ $\exists \delta \gt 0 : f$ is non-decreasing on $(x-\delta,x+\delta)$(I call that statement A). I need to prove that $f$ is non-decreasing on ...
3
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0answers
29 views

An inequality for a maximal function on an $n$-ball.

We have $Mf(x) = \sup_{r>0} \frac{c_n}{r^n} \int_{|y|\le r} |f(x-y)| dy$ the maximal function, where $r^n/c_n$ is the volume of the n-dimensional ball of radius $r$, $|y|\le r$. I want to show ...
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4answers
52 views

Evaluate $\lim _{x\to \infty }\frac{1}{x}\int _0^x\cos\left(t\right)dt\:$

I think that $\lim\limits_{x\to \infty }\frac{1}{x}\int _0^x\cos\left(t\right)dt\:$ is divergent, I can prove with taylor series?
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1answer
21 views

Tietze Extension Theorem ,,

If we have X a normal space, C a closed subspace of X, Y a completely regular space, and $f:C \rightarrow Y$ a continuous function. How do we show that f has a continuous extension $F: X \rightarrow ...
0
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0answers
45 views

If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ . [on hold]

I have tried a lot to solve this problem but am getting nowhere. Could someone please show me how it's done. Thanks. Note: $x_n$ is a sequence which is not necessarily convergent.
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0answers
14 views

Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
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0answers
12 views

Convergence in measure instead of almost everywhere convergence in DCT [duplicate]

Let $(X, M ,\mu)$ be an arbitrary measure space. Dominated convergence theorem requires some sequence of integrable functions to converges to some function f almost everywhere. However, if $f^n$ ...
0
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0answers
14 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
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1answer
36 views

Evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ and a big mistake in the book

We have to evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ where $f\left(t\right)=\frac{1}{\left(1+t^2\right)\left(1+t^3\right)}$. In my book they say that $\int ...
2
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2answers
36 views

Having trouble calculating $f_{xx}$ of a “variable-heavy” quotient.

Let $$ f(x,y) = \begin{cases} xy \frac{x^2 - y^2}{x^2 + y^2}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $$ Compute $f_x (0,0)$, $f_y (0,0)$, $f_{xx} (0,0)$, $f_{xy} (0,0)$, and ...
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0answers
7 views

Inferring Probabilities from relative frequencies

I have an question concerning the converse strong law of large numbers By the Converse Strong Law of large numbers, i mean the general principle (2) which is the converse of the standard strong law ...
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2answers
33 views

Give an example of a divergent and a convergent series such that the following holds: [on hold]

I'm having trouble with this: I need to find an example of a divergent series $\sum_{n=1}^\infty a_n$ of positive numbers $a_n$ such that $lim_{n \rightarrow \infty }$ $a_{n + 1}/a_n$ = $lim_{n ...
2
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2answers
59 views

Lebesgue measure of graph of $\sin{\frac{1}{x}}$ on $[0,1]$

I am working on something and read that measure of graph of a continuous function on compact sets is zero. Now, I tried to do it for non continuous functions but the set of discontinuities have ...
1
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0answers
34 views

Equivalence of 2 definitions of Differentiability

Let $X,Y$ be Banach spaces. I would like to prove the equivalence of the following definitions of differentiability. There is a map $\Delta : X \to L(X,Y)$ continuous at $a$, s.t. ...
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1answer
15 views

Pointwise Limits of functions

So the definition of a pointwise limit of a sequence of functions $f_{n}$ is $\lim_{n \rightarrow \infty} f_{n} = f$ if and only if $\lim_{n \rightarrow \infty} f_{n}(x) = f(x) \forall x$ in the ...
0
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0answers
16 views

Hypothesis needed for existence of an interval without a function zero

While studying ODE I thought of the following problem: Let $f:A\subset\mathbb{R}\to\mathbb{R}$ and $x_0\in A$ such that $f(x_0)=0$. What properties should have $f$ so as to allow us to conclude that ...
0
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0answers
39 views

If $f_n(x)$ are continuous functions from $[0,1]$ to $[0,1]$, and $f_n(x)\to f(x)$ as $n\to\infty$, then which of the following is true?

If $f_n(x)$ are continuous functions from $[0,1]$ to $[0,1]$, and $f_n(x)\to f(x)$ as $n\to\infty$, then which of the following is true ? (a) $f_n(x)$ converges to f(x) uniformly on $[0,1]$ (b) ...
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1answer
8 views

Example of a smooth $f$ such that $\sup_{t \in [0,1]}(f(t)/M - t)$ is not attained at $t = 0$

Let $f: [0, 1] \rightarrow \mathbb{R}$ be a non-negative smooth function which is not identically zero. Let $M := \sup_{t \in [0, 1]} f(t)$. Is there an example of an $f$ such that $$\sup_{t \in [0, ...
4
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2answers
167 views

One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
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1answer
24 views

Prove this function is uniformly continuous by verifying the $\epsilon$-$\delta$ property?

$f(x) = \frac{5x}{2x-1}$ on $[1,\infty)$ Here's what I've worked through so far: $$|f(x) - f(y)| = \left|\frac{5x}{2x-1} - \frac{5y}{2y-1}\right| = \left|\frac{5y-5x}{(2x-1)(2y-1)}\right| ...
2
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1answer
37 views

Do there exists continuous functions on compact sets with infinite length?

Is it possible to construct a continuous function from $[0,1] \to \mathbb{R}$ whose length is infinite?
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1answer
29 views

A characterization of Bessel sequences in a Hilbert space

I've shown that if for a sequence $\{f_{n}\}_{n=1}^{\infty}$ in a Hilbert space $H$ we have $$\sum_{n=1}^{\infty}|\langle f,f_n\rangle|^{2}< \infty$$ for all $f\in H$ (i.e., it is a Bessel ...
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3answers
59 views

Compact subset of $\mathbb R$ whose Lebesgue measure is non-zero

Let $\mathbb R$ be the field of real numbers, $\mu$ the Lebesgue measure on it. Let $K$ be a compact subset of $\mathbb R$. Is the following assertion true? If $\mu(K) \gt 0$, then the interior ...
1
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1answer
36 views

Convergence of $\int_0^1 \frac{\ln(1-x)\sqrt{x-x^2}}{\sin(\pi x)} \, dx$

I have difficulties with convergence of this integral: $$\int_0^1 \frac{\ln(1-x) \sqrt{x-x^2}}{\sin(\pi x)} \, dx$$ I found similar problem here Covergence of integral but I don't get the solution ...
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0answers
76 views

Exist the proof of Goldbach's Conjecture… is it correct? [on hold]

Every even integer > 2 is the sum of two prime numbers & equivalent Each odd integer > 5 is the sum of three prime numbers USING THE SIEVE OF ERATOSTHENES ...
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0answers
28 views

Some detail in the proof of the mean value formulae for harmonic functions

Let $B_r(x_0)$ and $\overline{B}_r(x_0)$ be the open and closed ball in $\mathbb{R}^n$, respectively $u\in C^2(B_r(x_0))$ and $\rho\in (0,r)$ $\lambda_n$ be the Lebesgue-measure on the ...
1
vote
2answers
60 views

Real Analysis: Continuity

$f(x)=\left\{ x^2+x, x \in \Bbb Q\right\}, f(x)=\left\{ x^3 + 1, x \notin \Bbb Q \right\}$ I want to prove that $f$ is discontinuous at $x \ne 1$. What I have so far is: Fix $\delta > 0$. We ...
2
votes
2answers
26 views

An open and closed ball in the discrete space

let $(X,d)$ be a metric space. I am trying to find what an open and closed ball looks like in the discrete space, i.e. when $d(x,y) = 0$ for $x = y$ and $1$ otherwise. Just considering the open ball ...
4
votes
4answers
84 views

How to evaluate $\lim _{n\to \infty }\:\int _{1/(n+1)}^{1/n}\:\frac{\sin\left(x\right)}{x^3}\:dx$?

We have to evaluate the following limit: $$\lim _{n\to \infty }\:\int _{\frac{1}{n+1}}^{\frac{1}{n}}\:\frac{\sin\left(x\right)}{x^3}\:dx,\:n\in \mathbb{N}$$ First step I wrote that $\int ...
3
votes
1answer
23 views

Fourier series and evaluation of another series

I was given to expand in a Fourier series the function $f(x)=|x|, \; x \in [-\pi, \pi]$. The Fourier series is quite known and I had done the calculations and I ended up to the formula: ...
0
votes
0answers
11 views

Continuity of integral from x to x+1 of Lp function

For $1 \le p < \infty$ and $f \in L^p({\bf R})$ define $g(x) = \int_x^{x+1} f(t) dt$. How do I shew that $g$ is continuous? In the case $p = 1$, we have $|g(x) - g(y)| \le \int_{y}^{x} |f(t)| dt ...
0
votes
1answer
16 views

Fourier transform of Schwartz functions

I am stuck with the following question: Suppose $f \in \mathcal{S}(\mathbb{R})$ satisfies $\widehat{f}(\xi)=0$ for $|\xi|<1.$ Prove that there exists $g \in \mathcal{S}(\mathbb{R})$ such that ...