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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2answers
36 views

Prove that if $A$ is both open and closed, $A=\mathbb R$.

Suppose $A$ is a non-empty subset of $\mathbb R$. Prove that if $A$ is both open and closed, $A=\mathbb R$. I think I'm supposed to assume that $A$ is not equal to $\mathbb R$ and derive a ...
2
votes
2answers
22 views

Finding the radius of convergence for the given series.

How to find out the radius of convergence for $\sum x^{n!}$? I tried the ratio and the root test. But no luck. Kindly help !
2
votes
1answer
19 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
1
vote
1answer
13 views

Proving Weierstrass Approximation Theorem on just bounded sets in $\mathbb R$

Suppose $f$ is a continuous function on $\mathbb R$. Show that we can approximate $f$ uniformly by a sequence of polynomials on any bounded subset of $\mathbb R$. My attempt is as follows: ...
2
votes
0answers
7 views

Function of mean square continuous process

I have been asked to prove that, if $\{X_t\}$ is a ($n$-dimensional) mean square continuous process and $f:\mathbb{R}^n \rightarrow \mathbb{R}^d$ is a Lipschitz function, the process $\{f(X_t)\}$ is ...
0
votes
2answers
32 views

a discontinuous function the square of which is continuous

give an example of a discontinuous function the square of which is continuous. The domain is $[0,1]$. I tried to use the indicator function of rationals, but its square is not continuous. EDIT:I am ...
3
votes
1answer
19 views

Heat kernel properties

I'm having problem with the heat equation in $\mathbb{R}^n$; specifically in proving the following: let $f\in L^1(\mathbb{R}^n)$ and: $$ u(x,t)=H_{\sqrt{t}}\star f(x)=(4\pi ...
1
vote
1answer
30 views

Prove $f'''(x) \geq 3$ for some $x \in (-2,2)$, if $f$ is cont on $[-2,2]$ and three times differentiable in $(-2,2)$ & $f(2)=-f(-2)=4$ & $f'(0)=0$

Prove that there exists a $x \in (-2,2)$ such that $f'''(x) \geq 3$, if $f$ is cont on $[-2,2]$ and three times differtiable in $(-2,2)$ with values $f(2)=-f(-2)=4$ & $f'(0)=0$. How do I handle ...
0
votes
3answers
26 views

Find some n such that $|s-s_n|< 10^{-3}$

Consider the series $\sum_{n=1}^\infty \frac{1}{n^2}$. Let $s_n$ be the $n$th of the series and $s$ be the sum of the series. Find some $n$ such that$$|s-s_n|< 10^{-3}$$ Can someone please ...
0
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2answers
35 views

For a normed vector space, prove, without the Hahn-Banach Theorem, that if $ L(x) = 0 $ for every continuous linear functional $ L $, then $ x = 0 $.

Let $E$ be a normed space. Is it true that if $L(x_0) = 0$ for every $L \in E'$ then $x_0=0$? One way to prove it is to consider $L \in E'$ s.t. $\|L\|= 1 $ and $L(x_0)= \|x_0\|$ (Existence of such ...
1
vote
1answer
20 views

Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
0
votes
1answer
24 views

What is the difference between uniform convergence and dominate convergence theorem?

I saw that both have aim to change limit with integral... that's the part that interests me most. I saw in some cases where we couldn't use uniform convergence, we use dominate convergence theorem to ...
-1
votes
0answers
21 views

Validity of Implicit Function Theorem

In essence, IFT says: If f(x,y) is C' and f(a,b)=0, then f(x,y)=0 is an identity (f(g(y),y)=0). But proof depends on f(x,y)=0 in a neighborhood of b. How do you know this? If you assume f(x,y)=0, ...
1
vote
1answer
36 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
1
vote
4answers
28 views

Find $\sup_{x\in[0,1]} \frac{x}{x^2+n^2+1}$

We have $f_n:[0,1]\to \mathbb{R},\:f_n(x)=\frac{x}{x^2+n^2+1}$ and we need to prove that is uniform convergence using formula: $\lim _{n\to \infty } \sup_{x\in[0,1]} |f_n(x)-f(x)| =0$ First ...
0
votes
0answers
21 views

Why $\int_{\mathbb R^n}e^{-\sum_{i=1}^n x_i^2} \, dx_1\cdots dx_n=\omega_{n-1}\int_0^\infty e^{-r^2}r^{n-1} \, dr$

Let $\omega_{n-1}$ the area of the surface of the sphere in $\mathbb R^n$, i.e. the surface of the set $\mathcal S=\left\{(x_1,\ldots,x_n)\mid \sum_{i=1}^1 x_i=1\right\}.$ We know that the area of ...
0
votes
2answers
54 views

Convergence of $\sum_{n=1}^\infty (2n^{10}+4n^5+1)/(4n^{15}+4n^{12}+5)$

Test whether the following series converges: $$\sum_{n=1}^{\infty}\frac{2\cdot n^{10}+4\cdot n^5+1}{4\cdot n^{15}+4\cdot n^{12}+5}$$ $1.$ I know that it does not make sense to use the root ...
0
votes
0answers
31 views

Conditional expectation over a convex set

Let $\boldsymbol{X}$ be an $\mathbb{R}^d$-valued absolutely continuous and integrable random vector. Further, let the cdf $F$ of $\boldsymbol{X}$ be strictly increasing in each component on ...
1
vote
1answer
33 views

Characterization of normed space

Let $\mathbb R^n$ be vector space over $\mathbb R$. Then we know that all norms over $\mathbb R^n$ are homeomorphism. Is it true for $\mathbb Q^2$ over $\mathbb Q$ For instance are the Euclidean norm ...
0
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0answers
12 views

“Transference” Argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
2
votes
1answer
23 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
0
votes
0answers
11 views

Commutation of Convolution, Restriction and Differentiation

Let $B$ be the open unit ball in $\mathbb R^n$ centered at zero and let $K=\bar{B}\cap (\mathbb R^{n-1}\times\{0\})$. Suppose you are given $u\in C^{1,\alpha}(B)$ such that $u|_K=f\in C^2(K)$. For a ...
2
votes
2answers
40 views

Evaluate if $f_{_n}$ converge uniformly or not

We have $f_n:[1,2]\to \mathbb{R},\:f_n(x)=\frac{x^n}{x^n+1}$ and we have to see if the convergence is uniform or not. From what I understand we need to prove that $\lim _{n\to \infty } ...
2
votes
1answer
43 views

If $d(x_{n+1},x_n)<\frac{1}{n+1}$ then the sequence $\{x_n\}$ is a Cauchy sequence OR not?

Let , $(X,d)$ be a metric space and $\{x_n\}$ be a sequence in $X$. We have, $$d(x_{n+p},x_n)\le d(x_{n+p},x_{n+p-1})+...+d(x_{n+1},x_n)$$ $$\le \frac{1}{n+p}+...+\frac{1}{n+2}+\frac{1}{n+1}\to ...
1
vote
1answer
60 views

convergency of the sequence $x_n=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+…+(-1)^{n+1}\frac{1}{n}.$

Test the convergency of the sequence $\{x_n\}$ , where $$x_n=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+(-1)^{n+1}\frac{1}{n}.$$ I think the sequence $\{x_n\}$ is convergent. As, ...
2
votes
0answers
36 views

Diffeomorphism from disk to plane

I want to show that the disk $D = \{(x,y) \in \mathbb R^2 : x^2+y^2 < 1\}$, the open square $K = (-1, 1)^2$ and the whole plane $\mathbb R^2$ are all diffeomorphic to each other. Therefore I want ...
1
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0answers
22 views

Need help understanding this proof of a certain inequality of $L^p$ norms.

The following theorem and proof is lifted from Folland (Real Analysis: Modern Techniques and their Applications). I am having trouble understanding one single line of the proof: Theorem: Let $K$ be a ...
0
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0answers
27 views

Lagrange multiplier method

Question 1: Could somebody please refer me to an introduction to Lagrange multipliers which is easy to read but still in full generality? Question 2: I am interested in particular in the following ...
3
votes
1answer
28 views

Prove a function that is nonnegative on a set then has a Riemann Sum that is greater than zero

Hey I have been working with Riemann Integrals and I have never sees this anywhere, but it makes intuitive sense to me and I can't seem to prove it. If you have a continuous function that is real ...
2
votes
1answer
41 views

If a sequence $f(x_n)$ goes to its minimum, will $x_n$ go to the point at which $f$ achieve the minimum?

I have a continuous function $f$ that is defined on a compact set. And $f(x_0)$ is its minimum. If I have a sequence $x_n$ such that $f(x_n)\to f(x_0)$, how can I show that $x_n\to x_0$? I tried ...
0
votes
0answers
19 views

Prove $\int_a^bf(x)dx = \int_{a+c}^{b+c}f(x-c)dx$ [duplicate]

Here's the problem: Let $f:[a,b] \to \Bbb R$ and let $c \in \Bbb R$. Prove that if $\int_a^bf(x)dx$ exists then so does $\int_{a+c}^{b+c}f(x-c)dx$ and these two integrals are equal. I've been ...
2
votes
1answer
26 views

Order the domain so that function is monotonic

Let $f : \mathbb{R} \to \mathbb{R}$ be a measurable function. Is there a bijection $b: \mathbb{R} \to \mathbb{R}$ such that $f \circ b$ is monotonic?
0
votes
1answer
16 views

Local Approximation of Real Valued Functions

I'm unsure where to begin. Any guidance would be greatly appretiated. Suppose that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ has continuous second order partial derivatives, and at the origin ...
2
votes
1answer
27 views

For what complex values of $z$ does the series $\sum_{n=0}^\infty \frac{z^n}{\log(n)}$ converge or diverge?

I used the ratio test to find that it converges when $|z| < 1$ and diverges when $|z| > 1$, but I'm not sure how to proceed with the $|z| = 1$ case. Because $z^n$ is function that traces out the ...
0
votes
0answers
18 views

Understanding bounded variation

In my analysis course we are covering the topic of bounded variation fuctions and I am really having a very hard time trying to get the concept. My main problem is that I don't get how can a function ...
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votes
0answers
20 views

Basics of determining if limit exists [on hold]

Find the limit or determine that it does not exist $\lim_{x\rightarrow c}\sqrt{x}$, for $c\geq0$
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0answers
10 views

How do I show that for a multivariate function, Lipschitz continuity in each variable implies Lipschitz continuity for the whole function?

This is a follow up to this answer that I came across while researching this. As an example, I have a function $f: \mathbb{R}^2 \to \mathbb{R}$ that is Lipschitz continuous in both $x$ and $y$ ...
1
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2answers
17 views

Image of a convergent sequence in an increasing function

Suppose I have a function $f:[a,b) \to \mathbb{R}$ that is increasing on its domain and a sequence $a_n \subset [a,b)$ such that $a_n \to b$ and $f(a_n)\to l\in \mathbb{R}$. How would I go about ...
0
votes
1answer
16 views

When is a series of sums the sum of the series?

In general, if $\Sigma_n (a_n+b_n)$ converges, then it may not be that $\Sigma_n a_n$ and $\Sigma_n b_n$ converge; for example, consider $\Sigma_n (1/n-1/n)$. If instead we know $\Sigma_n a_n$ and ...
0
votes
1answer
23 views

Find a suitable counterexample?

Is the following statement true or false? If a sequence $(x_n)$ with an infinite range $\{ x_n : n \in \mathbb{N} \}$ has precisely one accumulation point, then $(x_n)$ converges. I know the ...
2
votes
1answer
15 views

Convergence in measure of product of convergent sequences

Let $(X,\Sigma,\mu)$ be a finite measurable space ($\mu(X)<\infty$). Suppose $f_n \xrightarrow{\mu} f$ and $g_n \xrightarrow{\mu} f$, prove that $f_ng_n \xrightarrow{\mu} fg$ I'll write what I ...
1
vote
1answer
27 views

If f is continuous and strictly increasing, then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing.

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $f:I\rightarrow \Bbb R$ be a function. Suppose that f is strictly monotone. If f is continuous and strictly increasing (or ...
2
votes
3answers
33 views

Real analysis: simple second order ODE

I'm studying real analysis at the moment (just covered the mean value theorem, constancy theorem, applications to DEs etc.) and have run across this question that I'm stuck on. Any help would be much ...
1
vote
3answers
26 views

Show that the closed ball is closed in $\mathbb{R}^p$

Let $r>0, p \in \mathbb{N}$ be given. Show in detail that the closed ball $\{ x \in \mathbb{R}^p : ||x|| \leq r \}$ is closed in $\mathbb{R}^p$. Let $A = \{ x \in \mathbb{R}^p : ||x|| \leq r ...
1
vote
3answers
43 views

How to express sum as triple summation

I am trying to express the following sequences as summations: $$ 1+2^2+3^2+4^4+5^4+6^4+7^4 $$ and $$ 1+(2+3)^2 + (4+5+6+7)^4 $$ as summations. I think they will likely be triple summations, so ...
1
vote
2answers
19 views

base b expansion of real numbers

This is a problem in Zygmund's analysis book. It is intuitively very straightforward. However, I could not give a rigorous proof. I hope someone can show me how to prove this rigorously. Problem: ...
3
votes
3answers
98 views

Evaluate $\lim _{n\to \infty }\int_1^2\:\frac{x^n}{x^n+1}dx$

We have $$I_n=\int _1^2\:\frac{x^n}{x^n+1}dx$$ and we need to find $\lim _{n\to \infty }I_n$. Have any ideea how we can evaluate this limit?
3
votes
1answer
39 views

What are the fixed points of $f_ c = c · \sin$ for $c > 1$?

I’m doing an exercise for a lecture on dynamical systems. We are asked to classify all bifurcations of the dynamical system $f_c = c·\sin$ for real $c > 0$. We are given that bifurcations of ...
4
votes
0answers
21 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
0
votes
0answers
31 views

showing a function is continuous

While working on analysis continuity, i came across this question.I have understood the concept of continuity but cant see how to answer this question by including epsilon delta. $$f(x) ...