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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3
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1answer
26 views

Find extrema of $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$

Let $\overline{B_1(0)}\subseteq\mathbb R^3$ be the closed unit-sphere and $a\in\mathbb R^3$. Find all extrema of the function $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$ depending on $a$. ...
0
votes
3answers
87 views

A diffeomorphism with negative Jacobian swaps the orientation?

Let C be a simple close oriented curve $C^1$ in $\mathbb{R}^2$ and let $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a diffeomorphism such that $\forall (x, y) \in C$ it holds that the determinant of the ...
1
vote
2answers
45 views

Real Analysis Uniform Continuity Question help

If $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R}$, then $f$ is bounded on $A$. How would i start my proof? Could anybody give me a hint? I
5
votes
1answer
93 views

Uniform convergence of $\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$

Can someone please verify my answers? Consider the series $$\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$$ Prove that the series converges uniformly on the bounded interval $[-M, ...
19
votes
5answers
503 views

$f^2+(1+f')^2\leq 1 \implies f=0$

Find all $f\in C^1(\mathbb R,\mathbb R)$ such that $f^2+(1+f')^2\leq 1$ It's quite likely the answer is $f=0$. Note that $|f|\leq 1$ and $-2\leq f'\leq 0$. Therefore $f$ is decreasing and ...
2
votes
1answer
43 views

Convergence of similar power series given a convergent series

Can someone verify this? Suppose that the series $$\sum\limits_{n=1}^\infty a_n x^n$$ has a radius of convergence $R$, where $0 < R < \infty$ (a) Find the radius of convergence of ...
1
vote
4answers
46 views

Radius of convergence and sum of the series

I have the series $$\sum_{n=0}^{+\infty}\frac{x^{4n}}{9^{n+1}}$$ I'm supposed to find the radius of convergence and sum this order. I have tried finding the radius by using $$ R ...
1
vote
0answers
18 views

Some questions on the existence of Right-sided Limit of a Decreasing Function

Let $f:(a,b)\rightarrow\mathbb{R}$ be a decreasing function, bounded below. Show that $\lim\limits_{x\downarrow a}f(x)$ exists. This is the proof of the exercise I have but I have some problems with ...
3
votes
3answers
210 views

Integral $\int_0^{\infty} \frac{x^{a-1}}{1+x} dx $ converges?

For what values ​​of $a \in \mathbb{R}$ the following integral converges? $$\int_0^{\infty} \frac{x^{a-1}}{1+x}\ dx $$ I tried to compute the integral but I stuck solving and then I tried to compare ...
0
votes
2answers
107 views

Prove or Disprove: There are bounded, countable subsets of R^2 that have positive area.

Recall that a bounded set S in $R^2$ has area iff the inner area of S equals the outer area of S (and consequently the area of S equals said inner/outer area). My instinct tells me the statement is ...
4
votes
6answers
584 views

Intro to Real Analysis

I am having trouble proving the following: if $a < b$, then $a < {a+b\over2} < b$. I started with the Trichotomy Property and getting to where $a^2>0$, but then I do not know where ...
6
votes
0answers
171 views

The elementary methods to compute $\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\quad;\quad\text{for}\, \alpha>0$

How to compute the following integral using elementary methods (high school methods). \begin{equation}\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\qquad;\qquad\text{for}\, ...
1
vote
1answer
105 views

Some exam question on power series convergence

I provide my solution to the problem and wonder if I was thinking in a correct way. Find the radius of convergence of $$\sum_{n=0}{1 \over 1+n3^n}z^n$$ and give with reasoning a point $z_0$ on the ...
42
votes
3answers
956 views
0
votes
1answer
42 views

Norm of a vector is determined by evaluation of linear functionals on it: can this be proved without the Hahn-Banach theorem?

Let $V$ be a normed vector space over the field of real numbers, $\mathbb R$, and let $x_0 \in V$ be fixed. I know how to prove $$\|x_0\| = \sup_{f \in V^*, \|f\| = 1} |f(x_0)|$$ using the Hahn-Banach ...
1
vote
2answers
82 views

Differentiability of multivariate functions.

I would appreciate if someone could share some intuition as to the geometric meaning of differentiability condition of functions defined on $\mathbb{R}^n$. Such a function say ...
0
votes
1answer
49 views

I want to prove$ f(x) = \frac{x^4-9x+3}{x+1}$ is uniformly continuous on $[1,\infty)$

I am having difficulty getting $|f(x)-f(x_0)| < \epsilon$ Into the lipschitz inequality.
0
votes
1answer
47 views

Limit of a sequence defined by a recurrence relation

I have to find the limit of the sequence defined as follows: $u_0>0$ $u_{n+1} = \sqrt{u_n+\sqrt{u_{n-1} + \ldots + \sqrt{u_0}}}$ I really have no idea for that one... Not even how to start. ...
6
votes
3answers
383 views

A improper integral with Glaisher-Kinkelin constant

Show that : $$\int_0^\infty \frac{\text{e}^{-x}}{x^2} \left( \frac{1}{1-\text{e}^{-x}} - \frac{1}{x} - \frac{1}{2} \right)^2 \, \text{d}x = \frac{7}{36}-\ln A+\frac{\zeta \left( 3 \right)}{2\pi ^2}$$ ...
0
votes
1answer
62 views

Supremum / Infimum

Assume $f:\mathbb R \to \mathbb R$ is bounded and let $a< b$. Is is then true that $$ \sup_{x \in (a,b)} f(x) - \inf_{y \in (a,b)} f(y) = \sup_{x,y \in (a,b)} | f(x)-f(y)| $$ ?
0
votes
1answer
23 views

Determine the radius of convergence and sum the order

I need someone to check my work please. With regards to $A \in \mathbb{R}$, determine the radius of convergence and for $A = 0$, sum the order: $$\sum_{n = 0}^{+\infty} = \frac{A(-1)^n + ...
1
vote
2answers
32 views

$K$ compact $\subseteq A_1 \cup A_2$ open, then $\exists K_1 \subseteq A_1, K_2 \subseteq A_2$ compact s.t. $K = K_1 \cup K_2$

Let $K \subseteq \mathbb R^n$ be compact, $A_1, A_2$ open sets s.t. $K \subseteq A_1 \cup A_2$. Are there $K_1 \subseteq A_1, K_2 \subseteq A_2$ compact sets s.t. $K = K_1 \cup K_2$?
1
vote
1answer
58 views

Convergence of tricky series

I've stumbled upon a particularly unpleasant series, and I can't quite seem to crack it. $\sum\limits_{n=1}^{\infty}\dfrac{\ln(1+nx)}{n^{2}} $ I need to show uniform convergence on any interval of ...
0
votes
2answers
52 views

Integrating $\operatorname{sinc}(x)$ over an area

I was recently asked the following question in an exam and was unsure how to approach it: Evaluate the integral $$\int_{0}^{\pi}\int_{y}^{\pi}\frac{\sin(x)}{x}\:\mathrm{d}x\:\mathrm{d}y$$ I ...
3
votes
2answers
235 views

Determine a condition on $\vert{x-4}\vert$ that will assure…

This is a real analysis question regarding limits. The question is to determine a condition on $\vert{x-4}\vert$ that will assure that $\vert{\sqrt{x}-2}\vert<\frac{1}{100}$. My work: ...
0
votes
2answers
42 views

Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
0
votes
1answer
51 views

Real Analysis: Proving a theorem for limit of functions help please

I need to prove the equivalence relation: $\lim_{x \to c^+} f = L\Leftrightarrow $ for every sequence $(x_ n)$ that converges at $c$ s.t. $x_n\in A$ and $x_n>c$ $\forall n\in \mathbb{N}$, the ...
0
votes
2answers
58 views

Existence of $\delta$

Exercise Assume that $K$ and $A$ are disjoint nonempty subsets of $ \Bbb{R}^n$ with $K$ compact and $A$ closed. Prove by using wat we know about $d(x,A)$ that there exists $\delta >0$ such that ...
1
vote
3answers
33 views

Linearity of $D_{v}f(0)$

Exercise: Let $f:\Bbb{R^n}\rightarrow \Bbb{R}$ be homegeneous of degree 1, in the sense that $f(tx)=tf(x)$ for all $x\in\Bbb{R^n}$ and $t\in\Bbb{R}$. Show that $f$ has directional derivatives at 0 ...
-2
votes
1answer
120 views

Continuity of $|.|$ in $W^{1,p}_0$

please i dont understand this proof We suppose that $u\rightarrow u$ on $W^{1,p}_0$ i dont understand why we must use the weak compactness and the uniform convexity of $W^{1,p}_0$ ? Thank you
0
votes
0answers
37 views

Distribution function inequality for a transformed random variable ?!

I'm stuck with the following problem. Let $A:=\{g: g:(0,1)\to\mathbb{R}^+, non-decreasing\}$ and $U\sim U[0,1]$ be uniformly distributed on the interval $[0,1]$ on the probability space $(\Omega, ...
0
votes
1answer
73 views

An equality from Fritz John's paper

Prove: $\frac{\rho}{(4\pi)^2}\int_{|\xi|=1}d\omega_\xi\int_{|\eta|=1}f(x^0+r\xi+\rho\eta)d\omega_\eta=\int_{|r-\rho|}^{r+\rho}\frac{\lambda}{8\pi ...
0
votes
3answers
51 views

Series Expansion at n=infinity

Look at these two: $$a_n=\sum_{k=2}^n\dfrac1{k(k-1)}=\sum_{k=1}^n\left[\dfrac1{k-1}-\dfrac1k\right]=1-\dfrac1n<1. \\ \ \\ a_n=\sum_{k=0}^n ...
3
votes
2answers
35 views

Minimum of Integral, relation with area

Find the value of $a$ such that $F(a)$ is minimum, where $$F(a)=\int_{0}^{\pi/2} |\sin x - a\cos x| dx.$$ I want to differentiate the function but the absolute value prevented me from doing so... ...
2
votes
1answer
36 views

Show that $f^{-1}(K)$ is bounded

Exercise Let $f\colon\Bbb{R}^n\rightarrow\Bbb{R}^n$ be continuous and suppose there exists $k>0$ such that $||f(x)-f(x')|| \ge k||x-x'||$ for all $x,x'\in \Bbb{R}^n$. $i)$ Show that $f$ is ...
1
vote
0answers
84 views

Question about proof using epsilon delta for limits

When you use the epsilon delta method for proving limits and you face a problem of expressing $\delta$ as a function of $\epsilon$ only, I know that you have to put a restriction on $\vert{x-c}\vert$ ...
2
votes
2answers
58 views

$f'(x)=0 ,\forall x>0$ and $f'(x)=1 , \forall x \le 0$

Does there exist a function on the set of real numbers such that $f'(x)=0 ,\forall x>0$ and $f'(x)=1 , \forall x \le 0$ ?
4
votes
1answer
57 views

if a function $f$ is decreasing and the limit $\lim\limits_{x\to \infty} f(x)$ exists, then

Given that function $f$ is decreasing and the limit $\lim\limits_{x\to \infty} f(x)$ exists How can I prove $$\lim\limits_{x\to \infty} x\left(f(x)-f(x+1)\right)$$ exists? I applied monotone ...
2
votes
1answer
40 views

If $f+g$ and $f$ are differentiable at $a$, must $g$ be differentiable at $a$?

I've been asked to solve the following problem and I believe that I have gone about it correctly but I would appreciate a second look from someone with more experience. Question: If $f+g$ and $f$ are ...
0
votes
1answer
78 views

Proving $f(x) = f(0) + f'(0)x + \int_0^x (x-t) f''(t) dt$ for all x

Suppose f has a continuous second derivative. Prove $f(x) = f(0) + f'(0)x + \int_0^x (x-t) f''(t) dt$ for all x. Can someone check this for me? What I started with is that if we let $h(x) = \int_0^x ...
0
votes
3answers
148 views

Does (Riemann) integrability of a function on an interval imply its integrability on every subinterval?

For example, if $f$ is integrable on $[0,3]$, is it also integrable on $[1,2]$? I tried thinking of a counterexample but couldn't, since I've only learned what implies integrability but not what ...
2
votes
1answer
75 views

$S\subset \mathbb{R}^2$ with one and only one limit point in $\mathbb{R}^2$ such that no three points in $S$ are collinear

Question as in the title, but here it is re-typed just in case not all of the title is visible on your screen (you're welcome): I am interested if there is a set $S\subset \mathbb{R}^2$ with one and ...
0
votes
2answers
44 views

Pointwise and uniform convergence of this series

$$\sum_{n=1}^{\infty}\left(1- \frac{1}{2n}\right)^{-n^2}(x^2-1)^n$$ I've tried treating it as a power series centered around $x = 1$ and $x = -1$ and using root test I arrive to radius of convergence ...
1
vote
1answer
63 views

Growth of |logx| versus of 1/x

Do you think there is a number k s.t. $\int_{(0,\infty)} \frac{|log(x)|^{k}}{x}d\mu$ will converge,where $\mu$ is the Lebesgue measure? If you don't know ,can you at least give me some reference for ...
0
votes
1answer
45 views

Pointwise convergence of sequence of functions $f_n(x) = \sin(\frac{x}{n})$ where $f_n: \mathbb{R} \rightarrow \mathbb{R}$

I'm looking for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f_n \rightarrow f$ pointwise. I've tried graphing a few of the functions on WolframAlpha but it's not obvious what they ...
3
votes
2answers
113 views

what type of functions has $f(x+y) \geq f(x) + f(y)$

I was working on some $L^p$ inequalities and stumbled up on this. I know that $$f(x+y) \geq f(x) + f(y)$$ if $f$ is convex and monotone increasing. Does this hold for "only if"? And is there a name ...
1
vote
1answer
59 views

A particular quadratic minimization problem

Given $n^2$ constants $a_{11},a_{12},\ldots,a_{1n},a_{21},\ldots,a_{nn}$ and $n^2$ non-negative variables $x_{11},x_{12},\ldots,x_{1n},x_{21},\ldots,x_{nn}$. Find the minimum value of $$\sum_{i=1}^n ...
1
vote
1answer
37 views

differential inequality of continuous functions

Let $u:[0,+\infty)\to (0,+\infty)$ be a continuous function such that $\int_0^\infty u(x)dx<\infty$. Suppose there exist $a,b>0$ such that $\frac{du}{dx}\leq u(a+bu)$. Prove that ...
0
votes
0answers
57 views

Integrate over components of the unit vector

if I have a vector field$X: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ and $\phi:\mathbb{R}^3 \rightarrow\mathbb{R}$ such that $X(r, \theta, \phi) := \phi(r,\theta,\phi) e_j(\theta,\phi)$, where $e_j$ is ...
0
votes
1answer
65 views

What is the formal meaning of “determine” in Baby Rudin 2.40?

In Theorem 2.40, Rudin talks about a $k$-cell $I$ formed by the intervals $[a_1, b_1], \ldots, [a_k, b_k]$. We split each interval at its midpoint $c_j = \frac{a_j + b_j}{2}$ and end up with $2k$ ...