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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2
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1answer
28 views

complete metric space $X$ and nested sequence of closed sets $A_m \subset X$ where $\bigcap_{n=1}^\infty = \emptyset$ [duplicate]

What is an example of a complete metric space $X$ and a nested sequence of closed sets $A_m \subset X$ such that $\bigcap_{n=1}^\infty A_m = \emptyset$? My analysis professor mentioned this in office ...
1
vote
1answer
33 views

Riemann integrability of a square of a continuous function

Let, $f(x)$ be continuous in $[0,1]$ such that, $\int_{0}^{1}x^{n}f(x)dx=0$ for $n=0,1,2,3,...$. Then prove that, $\int_{0}^{1}f^{2}(x)dx=0$. First we apply $1^{st}$ M.V.T. of integral calculus & ...
3
votes
2answers
55 views

Find an $\epsilon$ such that the $\epsilon$ neighborhood of $\frac{1}{3}$ contains $\frac{1}{4}$ and $\frac{1}{2}$ but not $\frac{17}{30}$

I am self studying analysis and wrote a proof that is not confirmed by the text I am using to guide my study. I am hoping someone might help me comfirm/fix/improve this. The problem asks: ...
3
votes
2answers
48 views

Condition of the mean value theorem

The usual formulation of the mean value theorem in a real analysis course is something like this: Let $f\colon [a,b] \to \mathbb{R}$ be continous on $[a,b]$ and differentiable on $]a,b[$. Then there ...
-3
votes
2answers
20 views

Problem on CR inequality on finite sum [on hold]

Let $f$ be a function from {1,2,3,....,10} to R, s. t. $(\sum_{i=1}^{10}|f(i)|/2^i)^2=(\sum_{i=1}^{10} |f(i)|^2)(\sum_{i=1}^{10}1/4^i)$ mark the correct statement. A. there are uncountably ...
0
votes
0answers
44 views

Fourier series for logarithm of sine.

I looked up here: Fourier series of Log sine and Log cos I have modified the question: How can I derive the coefficient $a_n, b_n$ for $\log(\sin(x))$ in the fourier series representation? Also, I ...
1
vote
2answers
27 views

L2 norm and L1 norm inequality

In the vector space, we have the following inequality $$ ||x||_2 \leq ||x||_1 $$ where x is a vector. I am wondering that we have similar inequality for function's norm. L1 norm of function f is ...
0
votes
0answers
40 views

Construction of a strictly increasing, continuous function with zero derivative ae using the Cantor function

It is known that the Cantor function $\psi$ is a non-decreasing, continuous function with zero derivative ae. Let $n,k \in \mathbb N^+$ such that $k < 2^n$, and set: \begin{equation} f_{n,k}(x) = ...
0
votes
1answer
21 views

What does it mean to differentiate a map from $M_n$ to itself?

Here, $M_n$ is the space of real $n \times n$ matrices. This is in the context of differentiating functions $f: U \to \mathbb{R}^m$ (with $U \subseteq \mathbb{R}^n$ open), where the derivative at a ...
-1
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0answers
12 views

positive integrable part implies downside integrable

Let $A: M\rightarrow GL(d)$ measurable where $(M, \mathcal{B},\mu)$ is a probability space, then are equivalent: $$\log^+\Vert A^{\pm1}(x)\Vert\in L^1(\mu)\Leftrightarrow \log^-\Vert ...
2
votes
0answers
42 views

Convergence of Newton method under some assumptions

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 508 here) that if $x^\ast$ is the unique root of equation $$f(x)=0$$on interval $[a,b]$ and if the function has ...
0
votes
1answer
48 views

Problems on sequence and series of functions

Let $a_n$ be a sequence of real numbers. Which of the following is true? a. If $\sum a_n$ converges,then so does $\sum a_n ^4.$ b.If $\sum |a_n|$ converges,then so does $\sum a_n ^2.$ ...
2
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0answers
48 views

How to show that $f$ can only have at most one root in $(a,b)$ with these conditions?

Let $f: [a,b]\rightarrow\mathbb{R}$ be a differentiable function on $(a,b)$. Suppose $f$ has the following property: If for an $x \in (a,b)$, $f(x)=0$, then $f'(x)>0$. The excercise is to show, ...
1
vote
2answers
60 views

Showing that $f(x)=x^2$ for $x \in \mathbb{Q}$ and $f(x)=0$ for $x \not\in \mathbb{Q}$ is differentiable in $x=0$

I am supposed to show that $f(x) = x^2$ for $x$ in the rationals and $f(x) = 0$ for $x$ in the irrationals is differentiable at $x = 0$ and I am supposed to find the derivative of $f(x)$ at $x = 0$. ...
7
votes
1answer
82 views

How to derive this interesting identity for $\log(\sin(x))$ [duplicate]

I saw on SE that: $$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$ This is an extremely useful identity, as it helps solve: $$\int_{0}^{\pi} ...
0
votes
0answers
30 views

How to prove geometric mean is smaller than the arithmetic mean for a continuous distribution?

For discrete probability distribution, the geometric mean is defined as ${{\rm{E}}_{\rm{G}}}X = {\mu _G} = \sqrt[{\mathop \sum \limits_i {p_i}}]{{\mathop \prod \limits_i x_i^{{p_i}}}} = \mathop \prod ...
0
votes
1answer
26 views

Prelim problem in real analysis (any hints)

Find all constants $K > 0$ for which the following holds: If $(X,\Sigma,\mu)$ is any positive measure space and if $f:X\to \mathbb{R} $ is $\mu$ integrable satisfying $\left|\int_E ...
4
votes
0answers
48 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
4
votes
2answers
71 views

Does there exists a measure $\mu$ on $[0,1]$ such that $\int_0^1 p(x) \, d\mu(x) = p'(0)$ for every polynomial $p$

If this polynomial is at most degree $n$, I know this measure exists, but I am not sure that whether there exists $\mu$ for every polynomial?
2
votes
1answer
22 views

Extension of linear functional on $L^1$

Let $L^1([0,1])$ with the Lebsgue measure. Construct a bounded linear functional on some subspace of some $L^1([0,1])$ which has two distinct norm-preserving linear extensions to $L^1([0,1])$. For ...
1
vote
1answer
29 views

bounded intervals and partitions

Can you please check my proof? Question Let $I$ be a bounded interval of the form $I = (a, b)$ or $I= [a, b)$ for some real numbers $a< b$. Let $I_1, I_2, ..., I_n$ be a partition of $I$. ...
3
votes
1answer
40 views

If a sequence $f_n$ is bounded in $L^2$ and converges to zero a.e., then $f_n\to 0$ in $L^p$ for $0<p<2$

Let $M>0$, $\{f_n\}\subset L^2([0,1])$ such that $\int_0^1 |f_n|^2 dm\leq M$ and $f_n(x)\to 0$ as $n\to\infty$ almost everywhere, $m$ is Lebesgue measure. Show that for all $0<p<2$, ...
4
votes
1answer
27 views

For what $p$ does the surface of revolution for $x^p$ have finite surface area?

I am trying to investigate the surface of revolution of the $x^p$ functions, in the domain $[1,\infty)$ Using the formula for surface of revolution, $$A=2\pi\int_1^\infty x^p ...
4
votes
1answer
64 views

Use rolle's theorem to conclude $f(c)=f''(c)$

Let $a < b$ be two real numbers, and let $f : [a, b] \to \Bbb R$ be a differentiable function such that $f(a) = f(b)$ and $f'(a) = 0$. By applying Rolle’s theorem to the auxiliary function $h(x) = ...
-6
votes
2answers
41 views
1
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1answer
14 views

Understanding the proof of an Ergodic theorem for Markov chains

An ergodic theorem for Markov chains is as follows. If a Markov chain $(X_n)_{n \ge 0}$ is irreducible and has an invariant distribution $\pi$, then $$\frac{1}{n} \sum_{k=0}^{n-1} f(X_k) \to ...
-10
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0answers
33 views

$X\subset \mathbb R$ and $f,g:X\to \mathbb R$ be continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)=X$ [on hold]

Let $X\subset \mathbb R$ and $f,g:X\to \mathbb R$ be continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)=X$, Which of the following sets can not be equal to $X? $ A. ...
1
vote
2answers
44 views

If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ …

Suppose $f : [a,b] \to \mathbb{R}$ is continuous and $g \in \mathcal{R}[a,b]$ with $g(x) \ge 0$ for all $x \in [a,b]$. Show that there exists a $c \in [a,b]$ such that $$\int_a^b f(x)g(x) \, dx = ...
0
votes
2answers
35 views

Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
-7
votes
1answer
41 views

What can we say about the set $\cap U_i$? [on hold]

Let $U_1\supset U_2 \supset......$ be a decresing sequence of open sets in Euclidian 3 space $\mathbb R^3.$ What can we say about the set $\cap U_i$? A. It is infinite. B. It is ...
0
votes
1answer
11 views

If $|f_i(x_1) - f_i(x_2)| \leq a$ for all $f_i$'s, does $ | \min_i f_i(x_1) - \min_i f_i(x_2) | \leq a? $

Given a set $X$, $x_1, x_2 \in X$, and $f_i: X \to \mathbb R, i=1,\dots, m$. If $|f_i(x_1) - f_i(x_2)| \leq a$ for all $f_i, i=1,\dots, m$, does $$ | \min_i f_i(x_1) - \min_i f_i(x_2) | \leq a? $$
1
vote
0answers
30 views

sequence of linearly independent vectors

Lets say that we have I linerly independent vectors $\{v_1,v_2,...,v_I\}$. And lets say that we have a sequence of vectors $\{x^k\}^k$, where $x^k=\Sigma_Ic_i^kv_i$. Lets say that the sequence of ...
4
votes
1answer
63 views

Expressing “formally” $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series

I have to express $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series and give its interval of convergence. Knowing the binomial series is as follows this should be fairly easy: $$(1+x)^{\alpha}=\sum ...
-8
votes
0answers
21 views

Radon-Nikodym: Absolute Continuity and Mutual Singularity [on hold]

Find two measures $\mu$ and $\nu$ such that $\mu \hskip 0.4mm \not \hskip -0.4mm \ll \nu$, $\nu \hskip 0.4mm \not \hskip -0.4mm \ll \mu $, and $µ \hskip 0.4mm \not \hskip -0.4mm \perp ν$.
0
votes
1answer
23 views

Sequence of continuous functions in comple metric space.

Let $\{f_n\}$ be a sequence of continuous complex functions on a (nonempty) complete metric space $X$, such that $f(x)=\displaystyle \lim_{n\to\infty} f_n(x)$ exists for every $x\in X$. a) Prove that ...
7
votes
1answer
19 views

nonempty open set in normed space is connected iff each pair of points of the set can be joined by a polygon that lies wholly in the set

Let $E$ be a normed vector space. Let $x_1, \dots, x_m$ be points of $E$. Let $f(t) = (k-t)x_k + (t - k + 1) x_{k+1}$ for $k-1 \le t \le k$, $k = 1, 2, \dots, m-1$. The set $\{f(t)\text{ }|\text{ }0 ...
-2
votes
3answers
40 views

Is every point in $\mathbb R$ a cluster point of $\mathbb R$? [on hold]

Could someone please tell me if all of the points in $\mathbb R$ are cluster points for $\mathbb R$ or not?
2
votes
3answers
46 views

Function in $L^\infty$ is element of $L^2$?

Let $\mu$ positive measure and $f\in L^\infty(\mu)$. My question is: $f\in L^2(\mu)$? Thank you all.
1
vote
1answer
41 views

Geodesic equation

Assume that you have a parametrization of a surface $f:\Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3,(u,v) \mapsto f(u,v)$. Now if I have a curve defined by $g(t)=f(0,t)$. The geodesic ...
1
vote
0answers
33 views

Proving uniform continuity and uniform discontinuity

Could someone please explain to me how to show uniform continuity and not uniformly continuous for the following: $f(x) = \frac{1}{x^2}$ for $A = [1, \infty)$ show uniform continuity $f(x) = ...
1
vote
1answer
31 views

Is this true?: converging sequence question

Let X be a metric space, and let A ⊂ X. Suppose that {pn} is a sequence in A which converges to some point p ∈ X. True or false: i) p ∈ A′ (limit points of A) (ii) p ∈ closure(A) These are both true, ...
1
vote
0answers
23 views

Rayleigh quotient ($|r(q)-\lambda|=O(||q-x||^2)$ ?)

how to show $|r(q)-\lambda|=O(||q-x||^2)$ $r(q)=q^*Aq/(q^*q)$ and $\lambda$ is an eigenvalue, A is a symmetric matrix. x is the unit eigenvector corresponding to $\lambda$. and q is a unit vector. ...
1
vote
3answers
47 views

Need to show the following function is uniformly continuous on R

Could you please tell me how I am supposed to show that $f(x) = \dfrac{1}{(1+x^2)}$ is uniformly continuous in $\mathbb{R}$. I did some pre-calculation and found that $|f(x) - f(u)| < \epsilon$ if ...
1
vote
1answer
36 views

Show that $f(z):=\sum a_n (z-z_0)^n$ is continuous whenever $z$ is in disk of convergence.

Consider a power series $\sum a_n(z-z_0)^n$, and assume it has radius of convergence $r$. Then we know that $\forall z\in(z_0 -r,z_0 +r)$, this power series converges absolutely by root test. Thus we ...
2
votes
1answer
81 views

Prove $\lim_{n\to\infty}[(n^2+n)^{1/2}-n]= 1/2$

So I know that this for all epsilon>0, there exists an N such that for all n>N dp$ (p_n,\frac{1}{2})<epsilon $. But in this particular example I'm having difficulty finding the $N$ for which this ...
2
votes
1answer
20 views

lagrange interpolation, polynomial of degree $2n-1$

Let $a_1, \dots, a_n$ and $b_1, \dots, b_n$ be real numbers. How would I go about showing the following? If $x_1, \dots, x_n$ are distinct numbers, there is a polynomial function $f$ of degree $2n - ...
1
vote
3answers
50 views

Function converging in $L^1$

I've been having trouble with rigorously showing that on the interval $[0, 2\pi] $ the function $\sin^n(nx) \rightarrow 0$ in $L^1$ convergence. I've convinced myself that this is true intuitively by ...
0
votes
0answers
30 views

The “retraction” of $S\subseteq\Bbb R^2$ has rectifiable boundary

This is a continuation of the line of investigation of What's the most efficient way to mow a lawn? (although this question is self-contained). For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define ...
0
votes
1answer
67 views

Prove that limit goes to inf

Let $f:\mathbb R \to \mathbb R$ be such that $f(x), f'(x)$ and $f''(x)$ are all positive for each $x \in \mathbb R$. Apply the MVT to $f$ on each interval $[n,n+1]$ for $n=1,2, 3,\dots$ and show that ...
0
votes
3answers
63 views

How to show that a real continous function with image in the rationals is constant?

Can someone please explain to me how I am supposed to approach this question: If $f: [0,1] \to \mathbb{ R}$ is continuous, and has only rational values, then $f$ must be a constant.