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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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?
2
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0answers
16 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
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0answers
13 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 $L^\infty(0,T;H^1(\mathbb{R}^n))$?
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1answer
19 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 ...
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1answer
24 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
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2answers
27 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
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1answer
33 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
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1answer
23 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.
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0answers
15 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 ...
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5answers
94 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
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1answer
13 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
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0answers
10 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
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1answer
19 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 ...
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1answer
20 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 ...
10
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1answer
61 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
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2answers
26 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
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0answers
8 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 $\lambda[(A\cap V)\cup(B\cap V^c)]:=\mu(A)$ ...
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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
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2answers
26 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 ...
2
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6answers
52 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} ...
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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
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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
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1answer
20 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 ...
0
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1answer
20 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
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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
17 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
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0answers
17 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 $f$ ...
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1answer
37 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 ...
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1answer
19 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 ...
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0answers
17 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 ...
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1answer
23 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 : $$f(a,b,c)=f(|a|,|b|,|c|)$$ ...
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0answers
22 views

Given $\lim_n a_n = 0$, then $\lim_n r a_n = 0 \quad \forall r\in \mathbb{R}$?

Given $\lim_n a_n = 0$ then $\lim_n r a_n = 0 \quad \forall r\in \mathbb{R}$? I think it is true, then does it follow that for each $f\in C([0,1])$, we have $$\lim_n \int_0^1 f(x)\sin(nx) \; dx= ...
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2answers
22 views

What is the value of the following summation?

Compute $$\displaystyle\sum \limits_{n=0}^\infty (-1)^{n+1} \frac{1}{9^n(2n+2)}$$ I am given the fact that $$ \frac{1}{2}\ln(1+x^2) = \sum \limits_{n=0}^\infty (-1)^n\frac{x^{2n+2}}{2n+2} $$ ...
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0answers
21 views

show that if $f:[c,d] \rightarrow \mathbb {R}$ is continuous and $g:[a,b] \rightarrow [c,d]$ is Riemann integrable, then $f\circ g$ is also integrable

Please, give-me a hint to solve this problem: Show that if $f:[c,d] \rightarrow \mathbb {R}$ is continuous and $g:[a,b] \rightarrow [c,d]$ is Riemann integrable, then $f\circ g$ is also ...
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0answers
30 views

Proof of Bolzano-Weierstrass for functions over countable domains

Theorem A bounded sequence of functions defined over a countable domain has a convergent subsequence. The attempted proof uses Bolzano-Weierstrass for real sequences (nested bisections) proof and a ...
2
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1answer
47 views

If $\int_0^{x} g \leq \int_0^x f$ and $\phi$ is nonincreasing then $\int_0^{\infty} \phi g \leq \int_0^\infty \phi f$

Let $f, g$ be measurable real-valued functions on $[0, \infty)$, with $$\int_0^{x} g \leq \int_0^x f$$ for each $x$. Show that if $\phi: [0, \infty) \rightarrow [0, \infty)$ is nonincreasing, then ...
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1answer
38 views

If $\int_0^1 f(y)\sin(xy) dy = 0$ for every $x$, then $f = 0$ almost everywhere.

Can someone please give me a hint on this question, I have no idea where to start. Let $f \in L^p$ for some $1 \leq \infty$. Assume for all $x \in [0,1]$ that $$\int_0^1 f(y)\sin(xy) dy = 0$$ Show ...
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0answers
12 views

Can we find a natural number $m$ such that $[A^{c^{t}}]=[A^{c^{m}}]$?

Let $t$ be a non natural namber. Can we find a natural number $m≠t$ such that $$[A^{c^{t}}]=[A^{c^{m}}]$$ where $[x]$ is the integer part of $x$ (the floor function)? Here $A>1$ nad $c>2$ are ...
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1answer
18 views

Integral of absolute value

I have the following integral which I want to make sure to solve correctly and transparently: \begin{equation} \int_{\mathbb{R}}\|e^{ax}\|dx \end{equation} If I take cases I obtain: ...
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0answers
13 views

Find an Upper bound of absolute value (triangle equality application)

Given the functions f(x) and g(x), how can I find a bound for the absolute value \begin{equation} \|f(x)-g(x)-2\| \end{equation} is it correct to say $\|f(x)-g(x)-2\|\leq ...
1
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1answer
26 views

sequence of close and bounded sets in a prefect space

Suppose that$(E_n)$$_{n \in \mathbb N}$ be a sequence of closed and bounded sets in complete space $M$ such that $ E_{n+1} \subseteq E_n$ for all $ n \in\mathbb N$. If $\lim \operatorname{diam} E_n ...
1
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1answer
26 views

An element $u$ is an upper bound of $E$ if and only if $t>u$ implies $t\notin E$

Let $S$ be an ordered field and $S \supset E\neq \varnothing$. Then, the following are equivalent: $u \in S$ is an upper bound of $E$. $t \in S$ and $t > u$ implies $t \notin E $. My Try: ...
2
votes
0answers
23 views

On the greatest lower bound property

Proposition: Let $S$ be an ordered field and $S \supset E \neq \varnothing $. $E$ is bounded below. Then $ \inf E = - \sup ( - E ) $ Try: Write $- E = \{ -x : x \in E \} $ and let $l $ be a lower ...
0
votes
3answers
37 views

Compute two-dimensional integral over a region bounded by circular arcs

How to compute $$ \iint_{M}y\,{\rm d}x\,{\rm d}y $$ where $ \ M\equiv\left\{\,% \left(\, x,y\,\right)\ \mid\ y\ \geq\ 0\,,\quad x^{2} + y^{2}\ \leq\ 1\,,\quad \left(\, x - 1\,\right)^{2} + y^{2}\ ...
1
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1answer
47 views

Given $x,y\in\mathbb R$ is there a “formulaic” way to obtain a $q\in\mathbb Q$ with $a<q<b?$

Is there an assignment of reals $x,y$ to a rational number $q(x,y)$ for which $$\forall_{\mathbb R} x.\forall_{\mathbb R}(x<y).\left(x<q(x,y)<y\right)\hspace{.2cm}?$$ For computable reals, ...
3
votes
4answers
52 views

If $\dfrac{x_{n+1}}{x_n}$ converges to $l$, then $x_n$ converges to $0$

Suppose that $(x_n)$ is a sequence in $\Bbb{R}$ and that $$\lim_{n\to\infty}\dfrac{x_{n+1}}{x_n}=l$$ for some $l\in(-1,1)$. How do I show that $x_n\to 0$? For any $\epsilon>0$ we have an $N$ such ...
1
vote
1answer
53 views

Does $\sum\limits_{k=1}^\infty |\sin(ak)/k|$ converge?

Does $\sum_{k=1}^\infty |\sin(ak)/k|$ converge for all $0<a<\pi$? I do not think so since for $a=\pi /2$: $$\sum_{k=1}^\infty\left\vert\frac{\sin(ak)}{k}\right\vert=\sum_{k=0}^\infty ...
1
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2answers
36 views

Discontinuous everywhere but range is an interval

Does there exist a function which is discontinuous everywhere but range set is an interval.
0
votes
1answer
13 views

Another characterization of the supremum of a set

$u$ is an upper bound of a set $E \subset S$ if given any $\epsilon >0$, there is $\delta \in E $ such that $u - \epsilon < \delta$. PROBLEM: An upper bound $u$ of $E \subset S$ ($E \neq ...
3
votes
3answers
53 views

An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$

Title says most of it. Could you help me find an example? It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would ...