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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2
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1answer
72 views

union of two connect sets in particular case

Let $(X,d)$ is a metric space and $A,B \subset X$ are connected and $A \cap B = \emptyset$ and $A^- \cap B \neq \emptyset$ ($A^-$ is closure of $A$) now prove or disprove that $A\cup B$ is connected. ...
1
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1answer
39 views

Generalization of closed range theorem

For Hilbert spaces $X$ and $Y$, the closed range theorem is ok if the operator $T:X\rightarrow Y$ has a closed range. But do we in general have \begin{equation} Y=\overline{\operatorname{ran}(T)}\...
6
votes
2answers
274 views

Convergence of series of nested sequence

Let $a_n =\underbrace{\sin \left ( \sin \left ( \sin \cdots (\sin x) \cdots \right ) \right )}_{n \; \rm {times}}, \; \; x \in (0, \pi/2)$. Examine if the series: $$S=\sum_{n=1}^{\infty} a_n$$ ...
2
votes
1answer
74 views

$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

I am reading an article and there, author claim that $$‎L(.)=\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$$ is coercive if ‎‎$ ‎0\leq ‎‎...
3
votes
1answer
152 views

For which values does this series converge?

p and k are real numbers. For which values of p and k does the following double series converge $$\sum_{n,m=1}^\infty \frac{1}{n^p + m^k}$$ I am trying to find a better (and quicker) way to solve ...
1
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1answer
46 views

Is every real a finite linear combination of a proper subset of the reals with rational coefficients?

Wolfram Mathworld explains a Hamel basis as "a set of real numbers $\left\{U_\alpha\right\}$ such that every real number $\beta$ has a unique representation of the form: $$ \beta = \sum\limits_{i=1}^{...
0
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2answers
248 views

Proof that $\lim_{x\to 0} \sin(1/x)$ does not exist using contradiction

I am currently working through Apostol's Calculus, and I was hoping that someone could verify that the proof that I wrote for one of the problems actually proves the assertion. Prove that $\not\...
3
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2answers
99 views

Find an equivalent of this function,

a) $f$ continuous on $[0,1]$ such that $f(x)>0$. Find an equivalent of $$h(\epsilon) = \int_0^1 \frac {f(x)}{x^2 + \epsilon^2}dx$$ when $\epsilon$ goes to zero and when $\epsilon$ goes to ...
2
votes
3answers
129 views

upper bound on sum of exponential functions

Let $$\sum_{k=1}^{N} |a_k| = A$$ where $A$ is some constant. I am looking for an upper bound on the sum $$\sum_{k=1}^{N} e^{|a_k|},$$ independent of $N$, but may depend on $A$. That is, I am looking ...
0
votes
1answer
113 views

Apply Stone- Weierstrass Theorem [duplicate]

Suppose $f: [0,1]\to \mathbb R$ is continuous and $$\int_{0}^{1} f(x)e^{nx} \mathsf dx=0$$ for every $n$. Prove that $f(x)=0$ for all $x \in[0,1]$. Since $f$ is continuous on $[0,1]$, by Stone-...
2
votes
3answers
438 views

Bounded and closed but not compact in rational numbers

I'm sorry if topic repeated. I solved this problem. And I want to know is my solution true? Regard $\mathbb{Q}$, the set of all rational numbers, as a metric space, with $d(p,q)=|p-q|$. Let $E=\{p\...
2
votes
2answers
56 views

Is there an example of bounded non-measurable function?

My teacher's lecture note states bounded function defined on a measurable set is not necessarily measurable. Can anyone help give a concrete example? Thank you!
0
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4answers
302 views

Prove the laws of exponents by induction

We inductively define $a^1=a, a^{n+1}=a^n a$. I want to show that $a^{n+m}=a^n a^m$. By definition, this is true if $m=1$. Now for $m=2$, we have $$ \begin{align} a^{n+2} =& a^{(n+1)+1}\\ =&...
1
vote
1answer
39 views

If $E\subset[0,1]$, $|E|=0$ and $f(x)=x^3$, show $|f(E)|=0$, where $|E|$ denotes Lebesgue measure of $E$

The question is If $E\subset[0,1]$, $|E|=0$ and $f(x)=x^3$, show $|f(E)|=0$, where |E| denotes Lebesgue measure of $E$. Can anyone provide a hint on this? Thank you!
1
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1answer
87 views

$f:\mathbb R\to \mathbb R$ continuous, $f(f(0))=0$ so there exists $a \in \mathbb R$ such that $f(2a)=3a$

Let $f:\mathbb R\to \mathbb R$ continuous such that $f(f(0))=0$. Prove that there exists $a \in \mathbb R$ such that $f(2a)=3a$. Well, I figured that in such exercises, I should define a new function ...
0
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0answers
43 views

Find a countable family of closed intervals contained in $[0,1]$ such that the union covers $[0,1]$ but there is no finite subcover.

The question is Find a countable family of closed intervals contained in $[0,1]$ such that the union covers $[0,1]$ but there is no finite subcover. My solution is $\bigcup[\frac{1}{n},1]\...
1
vote
1answer
114 views

Should the empty set be called “half-open”? [closed]

Empty set is both open and closed in any metric space (also in any topological space). Consider $\mathbb{R}$ with usual metric. In this metric space, should we say that the empty set is half open?
0
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0answers
372 views

Product of two absolutely convergent series converges absolutely

From Baby Rudin, 3.13: Prove the Cauchy product of two absolutely convergent series is absolutely convergent. (I would like my proof reviewed because the solutions I looked at were much more ...
0
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1answer
46 views

Dunford Pettis Theorem

Suppose $L_1([0,1],\lambda)=L_1(\lambda)$ is the set of all $1$-integrable functions on $[0,1]$. $$S=\{(f_1,f_2)\in L_1^2(\lambda) |0\leq f_1+f_2\leq 1, a.e. \}$$ By Dunford Pettis theorem, we know ...
16
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1answer
135 views

Is there a continuous $f(x,y)$ which is not of the form $f(x,y) = g_1(x) h_1(y) + \dots + g_n(x) h_n(y)$

For a continuous function $f : [0,1]^2 \to \mathbb{R}$, let us say $f$ is a sum of products (SOP) if there exist an integer $n > 0$ and continuous functions $g_1, \dots, g_n, h_1, \dots, h_n : [0,1]...
0
votes
1answer
80 views

To show that function is Riemann Integrable on $[0,1]$

I am having difficulty in understanding this proof. Firstly why is the set $E$ defined? Why is specific value of $\delta$ chosen and about possibility of tags counting twice. Can anyone help me with ...
1
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1answer
24 views

convergence of disjoint subsequences

If $\{A_i\},i\in I$, is some partition of $\Bbb N$ such that $|A_i|=\aleph_0=|\Bbb N|,\forall i \in I$ , and all the sets $A_i$ are well ordered, then for a real sequence $\{x_n\}$ it is true that $...
1
vote
2answers
48 views

$f_n$ defined on $[-1,1]$ by $f_n(x) = |x|^{1+\frac1n}, \ x \in [-1,1]$ converges uniformly .

To show that the sequence of function $f_n$ defined on $[-1,1]$ by $f_n(x) = |x|^{1+\frac1n}, \ x \in [-1,1]$ converges uniformly to a function $f(x) = |x|, \ x \in [-1,1]$ and the sequence of ...
1
vote
3answers
56 views

$ \lim_{x \to \infty} x(\sqrt{2x^2+1}-x\sqrt{2})$ [closed]

Any ideas fot evaluating: $$ \lim_{x \to \infty} x(\sqrt{2x^2+1}-x\sqrt{2})$$ thanks.
0
votes
1answer
63 views

Baby Rudin 9.18 Implicit Function Theorem

Given f:Rn+m->Rn continuously differentiable and a=f(0,b). Prove z=f(0,y) in a neighborhood of y=b. BACKGROUND (and example of notation): In rough outline, Rudin and others prove the Implicit ...
0
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2answers
66 views

Looking for example of point wise convergent continuous bounded sequence of functions whose limit is neither continuous nor bounded

I am looking for a sequence of real valued functions $\{f_n(x)\}$ with domain some subset of $\mathbb R$ such that each $f_n$ is bounded , continuous and $f_n$ converges point-wise to some function $f$...
2
votes
2answers
58 views

Bounded function

If $f$ is a $C^{\infty}$ function $f: [0,T)\to \Bbb{R}$ continuous and bounded on $[0,T)$ and all its higher derivatives are also bounded on $[0,T)$. Then can we say that this function has a limit ...
7
votes
3answers
370 views

Infinite Integral of Trigonometric Functions

I am interested in finding the Integral: $I = \int\limits_{0}^{\infty} \sin x \,dx$. Clearly going the conventional way $I = -\cos (\infty) + \cos(0)$ will not lead to a definite answer. However I ...
1
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2answers
66 views

Understanding definition of Riemann Integral

I have read this definition first time today. As far as I can understand it, it seems to me that difference between Riemann sums and a number L can be made small by changing norm of partition. Does ...
2
votes
2answers
32 views

Uniform Convergence of $\sum_{n=0}^\infty (1-x^2)^2x^n$ on $[0,1]$; subsequent integral

Let $a_n = (1-x^2)^2 x^n$. Show that $\sum_{n=0}^\infty a_n$ converges uniformly on $[0,1]$ and deduce that $\int_0^1 \frac{(1-x^2)^2}{1-x} dx = \sum_{n=1}^\infty \frac{8}{n(n+2)(n+4)}$. Attempt: ...
4
votes
1answer
148 views

What's wrong with the argument $\sin^2{x}+\cos^2{x}=0$

I will just quote an argument given in the book Analysis 1 by Terence Tao. Start with the expression $\lim_{x→\infty}\sin(x)$ , make the change of variable $x=y+\pi$ and recall that $\sin(y+\pi)=-\...
0
votes
1answer
41 views

$f_n ^{'}$ is uniformly convergent but the seq $f_n$ is not uniformly convergent.

Let $f_n(x) = n + x/n , x \in \mathbb R$. Then the seq $f_n ^{'}$ is uniformly convergent but the seq $f_n$ is not uniformly convergent. We see that $f_n ^{'} = 1/n$ which is trivially uniformly ...
2
votes
2answers
106 views

Is every closed ball in metric space X bounded?

Assume that we have a metric space $X$ with Euclidean metric, and $r$ is a positive real number. Now let $B(a, r) = \{x \in X : d(a, x) \leq r\}$. Is $B(a, r)$ bounded? To me it seems that $0$ is a ...
1
vote
1answer
50 views

Trying to understand a power series example from Advanced Calculus by Taylor

Example 2 from 21.1 in the book, Find an expansion in powers of $x$ of the function $$ f(x) = \int_{0}^{1} \frac{1-e^{-tx}}{t}dt $$ and use it to calculate $f(1/2)$ approximately. I ...
0
votes
0answers
40 views

How to solve this equation for x

I'm trying to wrap my head around solving the following equation for $x$, but I can't find a way. Could anyone toss me a pointer? $$ x^{1-\alpha} + \beta x = C$$ for $\alpha > 0$, $\beta > 0$, ...
0
votes
2answers
57 views

Convergence of a product series with one $1/k$ factor

Let $\left( a_n \right)_n$ be a sequence such that $a_n < 1$, $a_n \rightarrow 0$. Prove or disprove (with a counter-example) that $$ \sum_{n=1}^{\infty} \frac{a_n}{n} < \infty.$$ Comments. If ...
1
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1answer
92 views

Question on uniformly convergent subsequence

Let $\{f_n\} \subset C([0,1])$ be a sequence of functions with $f_n(x) = x^n$ for $n = 1, 2, \ldots$ Prove that $\{f_n\}$ does not have any uniformly convergent subsequence. By (1) to prove that ...
3
votes
2answers
46 views

A proof regarding Lebesgue measure and a differentiable function

Could anyone kindly provide a hint on the following problem? My guess is to do some change of variables? Thank you! Let $f:\Bbb{R}\rightarrow\Bbb{R}$ be a continuously differentiable function. ...
4
votes
1answer
54 views

$\Bbb{Q} \bigcap [0,1]\subset \bigcup_{j=1}^NI_j$, show $\sum_{j=1}^N|I_j|\ge1$

The question is Let $\{I_1,I_2,...,I_n\}$ be a finite family of intervals in $\Bbb{R}$ such that $\Bbb{Q} \bigcap [0,1]\subset \bigcup_{j=1}^NI_j$, show $\sum_{j=1}^N|I_j|\ge1$ where $|I_j|$ is ...
4
votes
1answer
57 views

$\pi-\lambda$ Theorem to show measure giving interval lengths equivalent to Lebesgue on [0,1]

so I have been working on this problem and I want to make sure I am understanding the conclusion fully. So I have the following scenario: Not part of the actual question, but relevant. Consider ...
4
votes
3answers
104 views

Clarification about definition of bounded set of real numbers

I don't know I'm misinterpreting, but I'm a little confused about the definition of bounded. As far as I was aware, a bounded set of real numbers is one that is bounded both below and above. Therefore ...
0
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3answers
49 views

Uniform convergence and integration: $\lim_{n \to \infty} \int_{0}^{1} \frac{1}{1+x^2+\frac{x^4}{n}} dx=?$

For f$_n$(x)=$\frac{1}{1+x^2+\frac{x^4}{n}}$, we need to calculate $\lim_{n \to \infty} \int_{0}^{1} f_n(x) dx$ . I want to prove f$_n$ is Riemann integrable and f$_n$ uniformly converges to f, then ...
-1
votes
1answer
72 views

Show that $f(x) =\frac{1}{1+x^2}$ is Lipschitz continuous on $\mathbb{R}$? [duplicate]

Show that $$f(x) = \frac{1}{1+x^2}$$ is Lipschitz continuous on $\mathbb{R}$? I know $0<f(x)<1$ but $x$ and $y$ are any number in $\mathbb{R}$. How do you find a constant $c$ such that $$|f(x) -...
1
vote
1answer
51 views

To show function is not surjective

I am trouble understanding first three lines of proof . For onto we have to show that there exists some element in P(A) which doesnot have its preimage .But how did they have shown here
1
vote
1answer
62 views

Prove $\int_A {\cos nxdx} \to 0$ as $n \to \infty $ on any measurable subset $A$ of $[0,2\pi]$

Let $A$ be a measurable subset of $[0,2\pi]$, show that $\int_A {\cos nxdx} \to 0$ as $n \to \infty $. My attempt is that since $A$ is measurable, then $\forall \epsilon>0$ there exists an ...
1
vote
1answer
131 views

Example of differentiable function which has non-zero quadratic variation

The quadratic variation of a function $f : [a,b] \to \mathbb R$ is defined to be $$ \sup_P \sum_{i=1}^n (f(x_i) - f(x_{i-1}))^2 $$ where the supremum is taken over all partitions $P$ such that $a = ...
0
votes
1answer
30 views

Bound on the size of the solution to $u_{t} - \Delta u - u \leq 0$

Let $B$ denote the open unit ball in $\mathbb{R}^{d}$ and let $u$ be a smooth function such that $u_{t} - \Delta u - u \leq 0$ in $U_{T}:= B \times (0, T]$ and $u = 1$ on the parabolic boundary (that ...
1
vote
0answers
29 views

Let $f:[-2,5] \to \mathbb{R}$ integrable such as $f(x) \neq 0$ for all $x \in [-2,5]$, therefore $\frac{1}{f}$ is integrable in $[-2,5]$. T/F?

Let $f:[-2,5] \to \mathbb{R}$ integrable such as $f(x) \neq 0$ for all $x \in [-2,5]$, therefore $\frac{1}{f}$ is integrable in $[-2,5]$. I think that is true. Let's go for it. Let $g:I \to \mathbb{...
3
votes
5answers
214 views

Why can a closed, bounded interval be uncountable?

From what I have read, all finite sets are countable but not all countable sets are finite. As I understand it, Countably Finite --- a one to one map onto $\Bbb{N}$ with a limited number of members ...
6
votes
5answers
602 views

When a periodic function is squared (or cubed, and so on…) does it always lose its periodicity?

For instance $$\sin^{2}\left(-\frac{\pi}{6}\right) = \sin^{2}\left(\frac{\pi}{6}\right)$$ i.e., $\sin^2 (x)$ is an even function and loses the $2\pi$-periodicity of $\sin x $. Is this true in ...