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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2answers
29 views

Use the Mean Value Theorem to prove

Use the Mean Value Theorem to prove that $|\sin x - \sin y| \leq |x - y|$ for all $x,y \in \mathbb{R}$.
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0answers
34 views

Rational points on circle

I need help for the following questions. Give the necessary and sufficient condition for $r$ such that the circle $x^{2}+y^{2}=r^{2}$ passes the rational points. I know the obvious sufficient ...
4
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1answer
50 views

Proving $\lim_{x\to 0}f(x)=\infty$.

Suppose we have a function $f:[0,\infty)\to \mathbb{R}$ such that for every $N\in\Bbb{N}$ and every sequence of $\delta_n>0$ such that $\lim_{n\to\infty}\delta_n=0$, there exists $n$ for which ...
3
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2answers
63 views

computing an integration with a floor function

I am trying to compute $$\int_0^1 \left(\frac{1}{x} - \biggl\lfloor \frac{1}{x}\biggr\rfloor\right) dx$$ with no success. Any hints?
0
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1answer
37 views

Radius of convergence and sum of alternating series $1 - z + z^2 - z^3 + \ldots $

I have a (complex) function represented by the power series \begin{equation*} L(z) = z -\frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} \ldots \end{equation*} which I have tried to represent (perhaps ...
0
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1answer
26 views

Is a countable intersection of open sets in $\mathbb R$ Lebesgue measurable? [on hold]

If the answer is yes, how to prove that? Otherwise how to find a counterexample?
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0answers
32 views

a real analysis question,I need to prove whether two sequences are equidistributed or not,really need some help!

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
0
votes
1answer
45 views

Is an everywhere differentiable function locally Lipschitz?

If we have a differentiable function $ f:\mathbb{R}^n \to \mathbb{R}^n $, does it have to be locally Lipschitz? It's obviously true for continuously differentiable functions, but what happens without ...
4
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0answers
22 views

Difficult exercise on unicity of solutions for an IVP

Suppose $f$ and $g$ are continuous and $g$ is odd and strictly increasing function. I have to prove that the IVP $$y'=f(x)g(y)$$ $$y(0)=1$$ has a unique solution if and only if $$\lim \limits_{u \to ...
1
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1answer
14 views

Extending a convex function

Suppose $f:(a,b) \to \mathbb R$ is twice differentiable with the property that $c_1 \leq f''(x) \leq c_2$ for every $x \in (a,b)$, where $c_1, c_2$ are positive constants. Is it possible to extend $f$ ...
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1answer
38 views

Implicit Function Theorem Zero

The Implicit Function Theorem gives conditions under which f(p)=0 can be solved for some variables in terms of the rest. One of the conditions is f(p0)=0. How do you know a p0 exists and how do you ...
2
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1answer
53 views

Arithmetic mean of $L^2$ function is $L^2$

I have found the following problem, to which I do not find the solution: Consider $f(x), x > 0$ a function such as $$ \int_0^\infty f^2(x) dx < \infty $$ and let $g(x) = \frac 1x \int_0^x ...
2
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1answer
13 views

questions on polynomial Lagrange Interpolation of order $n$?

I ran in One Ex in my book when I‌ prepare for final exam on numerical method. how can help me how we solve such a problem? if $P(x)$ and $Q(x)$ be two polynomial Lagrange Interpolation of order $n$ ...
0
votes
1answer
26 views

Pointwise Limit of $f_n = \frac{x^n}{2+x^{2n}}$.

Let $f_n = \frac{x^n}{2+x^{2n}}$. What is the pointwise limit of this sequence of functions on $(0,1)$? We cannot say anything about $\frac{1}{x^n}$ since $|x| < 1$.
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1answer
51 views

How can I solve like this exercise

Let we have the following initial value problem : $$y'=f(x,y)=e^y$$ With the condition $y(0)=0$ Find the largest interval $|x| \le a $ makes the initial value problem has an unique solution ...
0
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1answer
24 views

$f:U \rightarrow \mathbb{R}$, $U$ is an open conected subset of $\mathbb{R}^n$ and $f \in C^1$ need to show that $f$ is $M$ Lipschitz on any compact

It is a more general form of the question here, only here $U$ is not a convex set but an open and connected subset of $\mathbb{R}^n$. I need to show that $f$ is $M$ Lipschitz on any compact $K \subset ...
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0answers
38 views

Showing that the sequence of functions is not Cauchy

I need to show that $ g_n(x)=x^{1/(2n-1)} $ is not a Cauchy sequence in $C[-1,1] $ w.r.t. supremum norm. I tried to find the maximum of the difference of $g_n$ and $g_m$ by just differentiating but ...
2
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0answers
22 views

Prove that the solution of an ODE can be prolonged to \infty

I need an help understanding some general techniques in ordinary differential equations. I've never attended a course on ODE, so I'm quite confused on the argument, but I'm trying to improve my ...
1
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1answer
23 views

Leibniz Rule question: Can this question be done as a function of single variables x?

We know for uniformly continuous functions in an interval [a.b] Leibniz rule applies $$\begin{aligned} \frac{d}{d\alpha}\int_{a(\alpha)}^{b(\alpha)} f(x,\alpha)\,dx &= \frac{d b(\alpha)}{d ...
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3answers
41 views

Prove that if $|x_{n+1} - x_n| \leq \frac{1}{n^2}$ for all terms of a sequence then the sequence is Cauchy.

Prove that if $|x_{n+1} - x_n| \leq \frac{1}{n^2}$ for all terms of a sequence, $n \in \mathbb{N}$ then the sequence is Cauchy. I have a proof but I'm not sure that it is correct, I would also like ...
2
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1answer
47 views

Where am I wrong in the following problem?

We have: $f:R\rightarrow R,\:\:f\left(t\right)=At^2-2Bt+C,\:where\:A=\int _1^2\:\frac{1}{x^2}dx,\:B=\int _1^2\:\frac{e^x}{x}dx,\:C=\int _1^2\:e^{2x}dx$ and we need to show that ...
1
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1answer
23 views

Integrability of dirichlet function in $\mathbb{R}^3$

Let $d: [0,1] \rightarrow \mathbb{R}$ be the Dirichlet function as follows: $$d(x) = \begin{cases} 1, & x \in \mathbb{Q} \\ 0, & x \in \mathbb{R} \backslash \mathbb{Q} ...
2
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3answers
47 views

Inequalities and Differentiation

Having become so accustomed to differentiation and integration being applied just like normal algebraic operators, I was somewhat suprised yesterday when I realized that $f(x) \geq g(x)$ does not ...
1
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2answers
19 views

Why is the gradient of the objective function in the Lagrange multiplier theorem not $= 0$?

A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ ...
0
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3answers
71 views

Does $\lim \frac{a_n}{e^{\delta n}}=0$ for every positive $\delta$ imply that $\lim \frac{a_n}{\sum\limits_{k=1}^n a_k} =0$

Let's say we have an increasing sequence $a_n$ such that $\underset{n\rightarrow\infty}{\lim} a_n=\infty$. Now it's fairly clear to me, though I haven't proven this yet, that: ...
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0answers
22 views

Find the range of values of p for which $\int_{0}^{e} x^{p} \log x dx $ converges. [on hold]

Find the range of values of p for which $\int_{0}^{e} x^{p} \log x dx $ converges.
2
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1answer
62 views

Showing a function $f:X \rightarrow Y$ is continuous

I am working through some practice questions, and I am not sure if I am on the right track with this one: Let $X = \cup_{n≥1}A_n$, be a topological space and assume that a map f : X → Y is such ...
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0answers
10 views

Identifying a subclass of the class of monotonic transformations

Let $u$ be a continuous function from $R$ to $R$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for greater than and ...
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4answers
110 views

Need help with proof of existence of $\sqrt{2}$

I am working my way through the proofs on this page. I am stuck on "4. The real number $\sqrt{2}$ exists." It begins: We will get $\sqrt{2}$ as the least upper bound of the set $A = \{q\in Q | q^2 ...
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1answer
19 views

Searching for a function where the inverse exists in a neighborhood of a point, but the Jacobian is zero.

I'm looking for a function f in $\mathbb{R}^2$ such that the inverse function therem at some point P = (x,y) does not give an answer of whether the function is invertable in some neighborhood of P, ...
2
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1answer
58 views

Prove $\lim\limits_{n \to \infty} \frac{n^2+1}{5n^2+n+1}=\frac{1}{5}$ directly from the definition of limit.

Prove $\lim\limits_{n \to \infty} \frac{n^2+1}{5n^2+n+1}=\frac{1}{5}$ directly from the definition of limit. So far ive done this: Proof: It must be shown that for any $\epsilon>0$, there ...
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1answer
22 views

Obtain the triangle inequality from the Minkowski inequality

I want to prove: $$\|f-h\|_p \leq \|f-g\|_p + \|g-h\|_p$$ Minkowski inequality: $$\|f+g\|_p \leq \|f\|_p + \|g\|_p$$ Is $\|f-g\|_p \leq \|f\|_p + \|g\|_p$? It looks like we have: $$\left(\int_a^b ...
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0answers
19 views

Tangent plane of a convex set.

Given a convex set $\Omega$ in $\mathbb{R}^n$ with smooth boundary, is $\Omega$ contained completely on one side of it's tangent plane at any point on the boundary? If so, how can we prove this? ...
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0answers
20 views

$c_{00}$ is a dense subset of $c_0$

I would like to show that $c_{00}$ is a dense subset of $c_0$. I am not sure if I am overly simplifying the argument or even making the right argument for that matter. proof: Suppose that $x \in ...
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0answers
11 views

Show that $v(E)=\text{sup}\sum_{j=1}^{n}|\mu(A_j)|$ is a measure.

Background A family $\textbf{X}$ of subsets of $X$ is a $\sigma$ algebra in case: $\phi, \mathbb{R} \in \textbf{X}$ $X \setminus A \in \textbf{X}$ if $A \in \textbf{X}$ If $(A_n) \in \textbf{X}$, ...
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0answers
17 views

On the hypothesis of the change of variable theorem

I´m studying the change of variable theorem for a function $f:\mathbb R^n \to \mathbb R$ and my teacher gave us the theorem as follows: Theorem: Let $f:A\subset \mathbb R^n \to \mathbb R$ be ...
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0answers
28 views

Show that the equation e ^(e sin (x)) = y is uniquely solvable [on hold]

a.) Show that the equation $e^{e \sin (x)} = y$ is uniquely solvable in a neighborhood $U_0$ of $x = 0$ for $y$ in a neighborhood $V_e$ of $e$. b.) Compute a first-order approximation to the exact ...
0
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0answers
16 views

continuity in $H_0^{1,2}$

Asuume that $u \in H_0^{1,2}(\Omega)$ and $f \in L^2(\Omega)$, and $\int_\Omega \nabla u \nabla v \, dx=-\int_\Omega fv \, dx$ holds for all $v\in C_0^1(\Omega)$. Show that $\int_\Omega \nabla u ...
3
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1answer
41 views

Contractions and finding Fixed Points

Define a map D on $C[0,1]$ by: $$D(f(x)) = \begin{cases} \frac 23 + \frac 13 f(3x), & \text{if }1 \le x \le \frac 13 , \\ (2+f(1)) (\frac 23 - x), & \text{if }\frac 13 \le x \le \frac 23, ...
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0answers
21 views

Prove f is (uniformly) Lipschitz continuous on the closed set D.

Let $D\subset \Bbb R$ be a bounded, open interval and suppose $f \in C(\overline{D} , \Bbb R)\cap C^1 (D, R)$ and $f′ \in C(D,\Bbb R)$. Prove $f$ is (uniformly) Lipschitz on $\overline{D}$, that is, ...
2
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0answers
50 views

Prove $ \sum \frac{cos(n)} { \sqrt n}$ is bounded [duplicate]

Prove $ \sum \frac{\cos(n)} { \sqrt n}$ is bounded. How can I prove $\sum \cos(n)$ is bounded. Can It be done using $\cos$ as real part of complex number as I showed in question ?
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1answer
16 views

Bounded set that is not closed nor compact

I am to find a set that is bounded but not closed nor compact. Here are my ideas. Please tell me if any of my logic is flawed. I thank you in advance. Consider the set $A = (0,1)$ where $A \subset ...
2
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5answers
90 views

How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$?

How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ? P.S: This is my method as I thought: $\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your ...
6
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2answers
46 views

Questions about the definition of convergence

I am having difficulty understanding the definition of convergence. I've been rereading and looking at examples during this past week and I haven't made any progress. Definition: We say that ...
2
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1answer
21 views

Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$

The definition I have of $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m)$ is that we require each component function to be in $L^p(\mathbb{R}^n)$. Is is true that $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m) ...
7
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1answer
64 views

Prove $\lim\limits_{n \to \infty} \frac{2n^2+2}{3n^3+1}=0$ directly from the definition of limit.

One of my homework questions is: Prove $\lim\limits_{n \to \infty} \frac{2n^2+2}{3n^3+1}=0$ directly from the definition of limit. In trying to follow: Prove that $\lim ...
1
vote
1answer
27 views

What are the images under $f$ of lines parallel to the coordinate axes?

Let $f=(f_1,f_2)$ be the mapping of $\mathbb{R^2}$ into $\mathbb{R^2}$ given by $$ f_1(x,y)=e^x \cos y,\quad f_2(x,y)=e^x \sin y.$$ What are the images under $f$ of lines parallel to the coordinate ...
2
votes
0answers
24 views

How much regularity is needed, anyway?

When doing real analysis, the difference between functions which are continuous and functions which are not is intuitive. The graph of the later may exhibit shearing, or extreme distortion (in higher ...
1
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1answer
83 views

Hints for evaluating $ \lim_{y \to +\infty}y \int_0^{+\infty}{e^{-x^2}\sin(2xy) dx}$ [on hold]

Please give me some hints for this limit. $ \lim_{y \to +\infty} y\int_0^{+\infty}{e^{-x^2}\sin(2xy) dx}$
2
votes
2answers
58 views

equation for the beta function

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function, proof the term: $(x + y)B(x + 1, y) = xB(x, y) \space\space \forall x, y > 0$ . Thanks in advance! ...