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, integration through the fundamental theorem of calculus.

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2
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1answer
14 views

Prove that {1/n}_n converges in the Sorgenfry topology.

Offer criticism, please! Consider $\{\frac{1}{n}\}_n=\{1,\frac{1}{2},\frac{1}{3},\cdots\}$. If $\{\frac{1}{n}\}_n$ converges, then $\{\frac{1}{n}\}_n \rightarrow 0$. By definition, $\forall (W \in ...
2
votes
1answer
29 views

Show that $\lbrace 1-\frac{1}{n} \rbrace_n$ does not converge in the Sorgenfry topology.

Consider $\{1-\frac{1}{n}\}_n=\{0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots\}$. If $\{1-\frac{1}{n}\}_n$ converges, then $\{1-\frac{1}{n}\}_n \rightarrow 1$. If it converges, then, by definition, ...
0
votes
1answer
22 views

$f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$

Assume that $f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$ for all $x\neq 0$ and $\alpha > 0$. Then I want to prove that $x \cdot ...
1
vote
1answer
28 views

If $f: U \rightarrow \mathbb{R}^n$ differentiable such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$, then $\det \mathbf{J}_f(x) \neq 0$

Let $f: U \rightarrow \mathbb{R}^n$ be a differentiable function on an open subset $U$ of $\mathbb{R}^n$. Assume that there exists $c>0$ such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$. ...
0
votes
1answer
18 views

Rational linear approximation in $C[0,1]$

I came across the following question in my homework and have been stuck for a long time Problem: Let $C_0$ be the countable subset of $C[0,1]$ consisting of all piecewise linear functions with ...
0
votes
1answer
30 views

Evaluate $\oint_C x^2 ds$

I'm trying to evaluate the line integral given by: $$\oint_C x^2 ds$$ Where $C$ is the curve of intersection of $x^2+y^2+z^2=25$ and $x+y+z=0$. Usually in these kind of problems one can use ...
6
votes
2answers
212 views

Is a differentiable function always integrable?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
0
votes
0answers
24 views

What does the positive minimal radius mean for an open cover?

The formal definition seems a bit confusing, any more words or pictures to explain it? Definition. Let $U$ be an open cover of $(M,d)$. We say that $U$ has positive minimal radius when there ...
2
votes
1answer
21 views

Proof about outer measure. For an interval $I$, $|I|_e=v(I)$?

My question is when proving $|I|_e \ge v(I)$, why cannot I conclude from $S={I_k}_{k=1}^\infty$ is a cover of $I$, then $v(I)\le \sigma(S)$, so $v(I)\le inf \sigma(S)=|I|_e$? Why do we need the ...
1
vote
2answers
12 views

Show every bounded infinite set has a maximum limit point and a minimum limit point.

Show every bounded infinite set has a maximum limit point and a minimum limit point. Here is my thought even if it is not formal Let $S$ be bounded and infinite set. Bolzano–Weierstrass theorem: ...
0
votes
2answers
28 views

$\sum\limits_{k=1}^{n^2}E(\sqrt{k})\quad n\in\mathbb{N}^{*}$

I found in my archives solution of this exercise Calculate $$\sum\limits_{k=1}^{n^2}E(\sqrt{k})\quad n\in\mathbb{N}^{*}$$ E represent the floor function Solution: they made Let ...
1
vote
1answer
14 views

Interpret the following sequence $X_n^{(k)} = \underset{1 \leq i_1 < \dots <i_k \leq n }{\sum} \; \xi_{i_1} \dots \xi_{i_k}$

I'm working on a problem in which I have the following set up. Let $\xi_1, \xi_2, \dots$ be independent random variables with $E[\xi_i] =0$ for all $i$. Then they define the following sequence $ ...
3
votes
1answer
54 views

If $\int f^n=\int f$then $f=\chi_E$

This is a problem in Real Analysis by Bruckner and Thomson: Let $f$ be a nonnegative lebesque integrable in the interval $[0,1]$, and suppose that for every integer $n=1,2,3,...$ ...
1
vote
3answers
74 views

Prove $f(x)\equiv C$

$f(x)\in C[a,b]$.For any $g(x) \in C[a,b]$ ,which has the property that $\int_a^b g(x) dx=0$,$\int_a^b f(x)g(x) dx=0$. Prove:$f(x)\equiv C$,$C$ is a constant. I haven't any ideas yet. I'm thinking ...
0
votes
2answers
19 views

Showing Convergence of Sequence using Cauchy Sequence

Show that $$\{S_n=\sum_{k=1}^n \frac{1}{k!}\}$$ is convergent by showing that $S_n$ is a Cauchy sequence. The hint given is $r!\geq2^{r-1}$ and $\sum_{r=1}^{\infty} \frac{1}{2^r}=2$ I know the ...
2
votes
3answers
45 views

If $f: \mathbb{R} \to \mathbb{R}$ is continuous then $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$

Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$. At first I thought this was quite ...
1
vote
1answer
15 views

some statements on sum of two subsets of plane. open, closed etc .

$W=\{(x,y)\in\mathbb{R}^2: x>0,y>0\}$ $X=\{(x,y)\in\mathbb{R}^2: x\in\mathbb{R},y=0\}$ $Y=\{(x,y)\in\mathbb{R}^2: xy=1\}$ $Z=\{(x,y)\in\mathbb{R}^2: |x|\le 1,|y|\le 1\}$ $W+X$ is open ...
0
votes
2answers
27 views

Are there extremely discontinuous functions?

Are there any functions $f:\Bbb R\to \Bbb R$ with the following property: For any $x_0\in \Bbb R$, any $\delta >0$ and any $\epsilon>0$ there is an $x$ with $|x-x_0|<\delta$ such that ...
0
votes
2answers
27 views

boundedness of the sequence $a_n=\frac{\sin (n)}{8+\sqrt{n}}$

How can i prove boundedness of the sequence $$a_n=\frac{\sin (n)}{8+\sqrt{n}}$$ without using its convergence to $0$? I know since it is convergent then it is bounded.
1
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0answers
36 views

An inequality concerning a particular function of two variables

How do you prove that ${x^y+y^x>1}$ if ${x,y>0}$ and both real? I saw a related problem on a Web puzzle page and it appears that the above statement is true when examined graphically in the unit ...
2
votes
0answers
24 views

On the supremum of the union of two bounded sets

Let $A,B$ be bounded subsets of an ordered set $S$. Then $A \cup B$ is bounded and $\sup( A \cup B) = \sup \{ \sup A, \sup B \} $. Attempt to solution: Let $x \in A \cup B$. Then $x \in A $ or $x ...
1
vote
1answer
33 views

Weak-* convergence in Sobolev spaces

Let's consider a sequence $\{f_n\}_n$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$. What does it mean that $\{f_n\}_n$ converges weakly-* in $L^\infty(0,T;H^1(\mathbb{R}^n))$?
0
votes
1answer
23 views

Some Dense subset of $M_2(\mathbb{R})$ with its usual topology?

The set of all invertible matrices i.e $GL_2(\mathbb{R})$ The set of all matrcies having both real eigen values. Having $Trace(A)=0$ $3$ is not dense set as It is closed set! $1$ Is dense. take ...
0
votes
1answer
28 views

denseness, connectedness and openness of a subset of $C[0,1]$

$X=C[0,1]$ with supnorm topology. Let $$S=\{f\in X:\int_{0}^{1} f(t)dt \ne 0\}$$ I need to know which of the following is/are true? $S$ is open $S$ is dense in $X$ $S$ is connected for the $1$ I ...
2
votes
2answers
35 views

a question about connected set, how to know whether A is connected or not?

In the Euclidean plane $R^2$,consider the subset $$ A=\{(x,y)\in \Bbb R^2|\text{Either $x$ or $y$, but not both, is a rational number}\} $$ Is $A$ connected? Is $\Bbb R^2$\A connected? I have ...
0
votes
1answer
53 views

Is the definition of continuity in analysis a particular case of topological continuity?

Take a constant function and remove an open interval from it: $$f(x)= 1, \text{if $x\in(-\infty,0]\cup[1,\infty)$ }$$ This function shouldn't be continuous because at $0$ no right limit of the ...
0
votes
1answer
25 views

Limits approaching from both sides go to infinity

Suppose that $\lim_{x \to a} f(x) = \infty$. Prove that we then have $\lim_{x \to a^+} f(x) = \infty$ and $\lim_{x \to a^-} f(x) = \infty$ from the definitions using epsilon-delta methods.
3
votes
0answers
27 views

Convergence in Measure, Different Definitions

Let $(X, \mu)$ be a measure space, $E \subseteq X$ measurable, and $f_n$ a sequence of measurable functions on $E$. If $f$ is another function on $E$, I have seen two definitions for what it means ...
4
votes
5answers
109 views

How to integrate $\int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x$?

I encountered this integral in the quantum field theory calculation. Can I do this: $$ \left. \int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x =x\ln\left(\, x\,\right)\right\vert_{0}^{1} ...
0
votes
1answer
17 views

A basic logical question on on sequence of functions

Let $f_n:[0,\infty)$ and $f:[0,\infty)$ be a sequence of functions such that for every finite $T$, $f_n:[0,T]\rightarrow f:[0,T]$ uniformly. But, it need not be true that $f_n:[0,\infty)\rightarrow ...
0
votes
0answers
11 views

If $a \frac{\partial}{\partial x} (f+g) = \sin(f-g)$ and $f= f_0 + af_1 + a^2 f_2 + a^3 f_3+…$, then finding $f_0, f_1, f_2$ and $f_3$

Let $g(x)$ be a smooth function and assume that for each $a$ in a neighborhood of $0$ there exists a function $f(x,g,a)$ which is smooth in $x$ such that $a \frac{\partial}{\partial x} (f+g) = ...
2
votes
1answer
33 views

If $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$, then $f$ is a diffeomorphism

Suppose that $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ is a differentiable function such that $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$. (Note that $\mathbf{J}_f$ is ...
-1
votes
1answer
26 views

Show that any bounded linear functional on a normed linear space is continuous

Show that any bounded linear functional on a normed linear space is continuous. Can we say that it is uniformly continuous ? Also, is it true, if we reverse the statement any continuous linear ...
12
votes
3answers
99 views

Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.

Let $C$ be the class of all real-valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1$. How to show that $$\inf_{f\in ...
3
votes
2answers
33 views

Finding a generalized form for this series

While i was just playing around with series i came across this one, $$ S = \sum_{k=1}^\infty[\frac{k}{k-\frac{1}{2}}+\frac{k-\frac{1}{2}}{k}-\frac{k+\frac{1}{2}}{k} - \frac{k}{k-\frac{1}{2}}] $$ ...
0
votes
1answer
12 views

Correctly defined measure

I need to show that the measure is unique (correct definition). Prove that the function $\lambda: \sigma(\mathcal{A}\cap V) \rightarrow [0,1]$ such that $\lambda[(A\cap V)\cup(B\cap V^c)]:=\mu(A)$ ...
0
votes
0answers
28 views

Show that $\int_{0}^{1} |g_{n} − g|d\mu \to 0$ as $n \to \infty$.

Let $\mu$ be Lebesgue measure on $[0, 1]$. Let $g : [0, 1] \to \mathbb{R}$ be Borel measurable, and set $g_{n}(x) =g(\frac{nx}{n+1})$. Assume that $g$ is bounded, and that $g$ is continuous at $x$ for ...
0
votes
2answers
28 views

Prove or disprove: for any two given functions, one must be upper bounding the other

$f \in O(g)$ definition:$$ \exists c \in \mathbb{R^+}, \exists B \in \mathbb{N}, \forall n \in \mathbb{N}, n \geq B \Rightarrow f(n) \leq cg(n) $$ $f \in \Omega(g)$ definition:: $$ \exists c \in ...
3
votes
6answers
58 views

Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.

Question: Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$. Attempt: Using L'Hopital's Rule, I have come to $$ \lim_{x \to 0} \frac{\cos(x)}{2x} - \lim_{x \to 0} ...
1
vote
1answer
14 views

calculate $\lim_{n\to\infty}\int_{[0,\infty)} \exp(-x)\sin(nx)\,\mathrm{d}\mathcal{L}^1(x)$

We've had the following Lebesgue-integral given: $$\int_{[0,\infty)} \exp(-x)\sin(nx)\,\mathrm{d}\mathcal{L}^1(x)$$ How can you show the convergence for $n\rightarrow\infty$? We've tried to use ...
3
votes
1answer
32 views

Absolute convergence of $ \sum a_nx_n $ implies absolute convergence of $ \sum a_n$

I'm trying to find a proof (or a conunter example, but I'm somehow convinced that the statement is true) for the following fact: $$ \forall_{(x_n)_{n=1}^{\infty} \lim{x_n} = 0 } ...
0
votes
1answer
21 views

Finding the limit of an exponential term minus a polynomial term

Find the limit for $$\lim_{x\to +\infty} e^x - x^2$$ The result should be $+\infty$. How do you do it? If you convert the equation to a quotient: $$\frac{1-e^{-x}x^2}{e^{-x}}$$ and apply ...
-1
votes
1answer
27 views

Behavier of function and its derivatives at infinty

If $\lim_{t\to \infty}(\phi(t))=0$ and $\lim_{t\to \infty}(\phi''(t))=0$ then can we say $$ \lim_{n\to \infty}(\phi'(t))=0$$ Can we have a $\phi(t)$ such that $\lim_{t\to \infty}(\phi(t))=0$ but ...
0
votes
1answer
18 views

Building an nth order ODE in Maple (or Matlab)

The question is simple: given a system of ODEs, how can one construct the equivalent nth order ODE in Maple? In my case I have $$ \begin{cases} y''(t)+x'(t)+x(t)=f(t)\\ y''(t)+z''(t)+z'(t)+z(t)=0\\ ...
3
votes
1answer
25 views

Directly proving continuous differentiability

Let us say that we want to prove that a function $f: I \to \mathbb{R}$ defined on an open interval $I$ is continuously differentiable on $I$. One way to do this is to establish that $f'(x)$ exists at ...
3
votes
0answers
21 views

Help with an Analysis problem involving isomorphisms and volumes.

I would like some help solving the following problem: Let $f: U \subseteq \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ be a class $C^{1}$ function. Suppose that for some $a \in U$ the derivative of $f$ ...
0
votes
1answer
41 views

What is the largest complete subspace of $(\mathbb{Q}, |\cdot|)$

For example $\left\{\frac{1}{n}\right\}\cup \{0\}$ is a complete subspace of $\mathbb{Q}$, but I am having trouble writing out the largest (in the sense of "$\subset$") complete subspace in ...
1
vote
1answer
20 views

a question about compact set, how to prove there exits f(y)=y [duplicate]

Let (X,p) be a compact metric space.Suppose that f X->X is a function such that, for all $x_1$,$x_2$ $\in$X, if $x_1\neq x_2$ then p(f($x_1$),f($x_2$))<$p($x1$,$x2$)$. Prove that there exits a ...
2
votes
1answer
49 views

Prove $f'(x) \geq x f(x)$ $\forall x \in \mathbb{R}$ $\implies$ $\exists k$ s.t. $ke^{x} \leq f(x)$ $\forall x \in \mathbb{R}$.

Question: Let $f$ be a differentiable function. Prove $f'(x) \geq x f(x)$ $\forall x \in \mathbb{R}$ $\implies$ $\exists k$ s.t. $ke^{x} \leq f(x)$ $\forall x \in \mathbb{R}$. I've made some ...
1
vote
1answer
25 views

function of three variable is even than $f(a,b,c)=f(|a|,|b|,|c|)$

I used in the proof of Hlawka's Inequality you can find the link here Hlawka's Inequality that's if i have function of three variable is even in each variable, so that : $$f(a,b,c)=f(|a|,|b|,|c|)$$ ...