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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4answers
204 views

Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.

The title says it. I thought of the following: we want $$\Bbb N = \dot {\bigcup_{n \geq 1} }A_n$$ We pick multiples of primes. I'll add $1$ in the first subset. For each set, we take multiples of some ...
0
votes
0answers
101 views

Can you find a formula for the supremum of an infinite collection of nonempty sets each of which is bounded above?

I believe the supremum of a collection of finite sets which are all nonempty and bounded above is simply the maximum of the set containing the supremums of each set, but can you find the supremum of a ...
1
vote
0answers
56 views

Application of the Hahn-Banach Theorem from Folland

I'm struggling to understand the following proof of the theorem stated in Folland (p. 159). 5.8 Thm Let $X$ be a normed vector space. If $M$ is a closed subspace of X and $x\in X\backslash ...
0
votes
1answer
38 views

Is this sequence of function uniform convergence

Let $C[0,1]=\{ f:[0,1] \to \mathbb{R} \mid f \text{ is continuous} \}$ equipped with sup norm $\lVert f \rVert_\infty = \sup_{x\in [0,1]} \lvert f(x) \rvert$. I don't know that a sequence function $...
0
votes
1answer
55 views

If $(x_n)_{n\in\mathbb{N}}$ and $(y_n)_{n\in\mathbb{N}}$ are convergent in $\mathbb{R}^m$ , then $\|x_n - y_n\|$ is convergent in $\mathbb{R}$

To demonstrate this proposition I use the following: Definition $\quad$ $(x_n)_{n\in\mathbb{N}}\subset \mathbb{R}^m$ converges to $x\in\mathbb{R}^m$ if and only if $\forall \varepsilon >0 $ there ...
0
votes
0answers
64 views

Showing $\exists$ unique $x>0$ such that $x^{2} = k$

I'm trying to work through this proof, and have hit a stumbling block. This is what I have so far: Let $0<k \in \mathbb{R}$, where $E = \{ x \in \mathbb{R} | x^{2} < k\}$. First, we must show ...
-1
votes
1answer
32 views

Find bounding functions by using Lagrange Mean Value Theorem .

Suppose that on the interval $[-2,4]$ the function is differentiable, $f(-2) = 1$ and $|f' (x) | \leq 5$. Find the bounding functions of $f$ on $[-2,4]$, using LMVT. I solved it as $$-5\leq \frac{f(x)-...
3
votes
5answers
1k views

A function that is not a derivative [closed]

If this is possible, could somebody give me an example of a function $f:[a,b]\to \Bbb{R}$ continuous in $[a,b]$ but that THERE IS NOT a function, namely G, such that its derivative is the funciont f? ...
1
vote
1answer
49 views

Elementary Measure theory problem, and graduate school inquiry .

I actually have one question, and one concern. Let $X$ be a metric space and let $\mu$ be a measure in $B(X)$. Prove that $\mu^*(E)=$inf $\mu(U)$, where $U$ is open and $E\subset(U)$ defines a ...
4
votes
2answers
134 views

Measure theory qualifying exam questions

The following question is a qualifying exam question, though I don't see how these two parts are related. (a)Let $(X, \mathcal F, \mu)$ be a measure space with $\mu(X)=1$ and suppose $F_1 , F_2, ......
1
vote
2answers
89 views

Convergence of a sequence of convolutions

Let $(a_n)$ be a sequence of real numbers such that $$ a_0>a_1>\cdots>0 $$ and $M:=\sum_{n=0}^\infty a_n<+\infty$. Denote $$ g_n=\frac{1}{a_n}\cdot 1_{[0,a_n]} $$ and define $$ ...
2
votes
2answers
129 views

Exercise: signed measures, total variation.

I have this exercise: Let $\nu_1$ and $\nu_2$ be finite signed measures on $(\Omega,\mathcal{A})$. Prove that: $|\nu_1+\nu_2|\le|\nu_1|+|\nu_2|$; , that is $|\nu_1+\nu_2|(A)\le|\nu_1|(...
1
vote
1answer
38 views

Product of limits of two sequences

Let $x_n$ be a sequence of positive numbers. Let $$s_n = \frac{x_1 + \ ... \ + x_n}{n}$$ $$t_n = \frac {\frac{1}{x_1} + \ ... \ + \frac{1}{x_n}}{n}$$ Prove that if $s_n \to S$ and $\ t_n \to T, \ ...
4
votes
1answer
60 views

Proving or disproving continuity of $f(x+a)=\frac{1+f(x)}{1-f(x)}$

Let $f$ be defined $\forall x\in\mathbb{R}$ and let $$ f(x+a)=\frac{1+f(x)}{1-f(x)} $$ hold for some $a\in\mathbb{R}^{+}$ and $\forall x\in\mathbb{R}$. Show that $f$ is periodic with a period of $4a$ ...
2
votes
2answers
48 views

“alternate definition” of continuity

I was trying to find a definition of the continuity of a function that fits better the unrirgorous definition that "it can be drawed with a pencil in a single stroke" and I came up with this: a ...
2
votes
1answer
47 views

Integral over a torus - change of variables

Heyyyy I would like to understand how to perform the variables change $\phi \rightarrow \psi$ in the following sum $$ \int_{\varphi ...
1
vote
0answers
19 views

Relation between periodic and continuous functions?

Is there any relation between them? I know that every periodic and continuous function is bounded and uniformly continuous. For counterexamples of periodic $\implies$ continuous or continuous $\...
3
votes
2answers
47 views

Iterative scheme involving cosine.

Define a sequence of real numbers $\{x_k : k \ge 0\}$ by $x_0 = 0$ and $x_k = \cos(x_{k-1})$. Does the limit $\lim_{k \to \infty} x_k = z$ exist?
0
votes
0answers
14 views

Euclidean regular hypersurfaces VS. Regular hypersurfaces in Heisenberg group

Let's fix the setting we are working in: let $\mathbb H^n$ be the Heisenberg group, we denote its points $P$ as $P=(x_1,\dots, x_n,y_1,\dots,y_n,t)$, the Lie algebra as $X_i=\frac{\partial}{\partial ...
1
vote
2answers
15 views

Determine the field properties that are satisfied by B, Is B a field?

Let B be the set of all irrational numbers together with the numbers 0, 1, and -1. Let addition and multiplication be defined on B in the same way they are defined for real numbers. Determine the ...
0
votes
0answers
35 views

show that any set that is open wrt metric D(x,y)=min{1,d(x,y)} is open wrt d

Q. Let (Y,d) be metric space. let D: Y x Y --> R be: D(x,y)=min{1,d(x,y)} show D is also a metric. show that any set that is open wrt D is open wrt d and vice versa. my attempt: i have done ...
4
votes
1answer
55 views

Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .

Prove that if $f$ is differentiable on $ [a,b]$ and if $ f(a)=f(b)=0$ then for any real $k$ there is an $ x \in (a,b) $such that $$kf(x)+f'(x)=0 $$ As all the conditions of Rolle's theorem are ...
0
votes
1answer
65 views

Problem from baby Rudin about completion of metric space

Let $X$ be a metric space and $C[X]$ the set of all Cauchy sequences in $X$. Call two $\{p_n\},\{q_n\}\in C[X]$ equivalent if $\lim \limits_{n\to \infty}d(p_n,q_n)=0$. This is an equivalence ...
8
votes
3answers
183 views

Computing $\lim_{A\to\infty} \frac{1}{A} \int\limits_1^A \! A^{\frac{1}{x}} \, \mathrm{d}x.$

On this year's IMC there was this problem: Compute $$ \lim_{A\to\infty} \frac{1}{A} \int\limits_1^A \! A^{\frac{1}{x}} \, \mathrm{d}x. $$ In addition to the two official solutions, I am curious as ...
1
vote
1answer
54 views

Bolzano-Weierstrass: “Neighbourhood of x contains an infinite number of members from a sequence”? Accumulation point?

I've not heard of these terms before, but was viewing a Bolzano-Weierstrass theorem proof that proved that the "Neighbourhood of x contains an infinite number of members from the sequence" in order to ...
2
votes
1answer
55 views

$H$ is Hilbert, countable basis. If $||x_n|| \to ||x||$, and $\langle x_n,y\rangle \to \langle x,y\rangle \forall y\in H$. Show $||x_n-x|| \to 0$

Problem Statement: Suppose $H$ is Hilbert, with a countable basis. If $||x_n|| \to ||x||$, and $\langle x_n,y\rangle \to \langle x,y\rangle$ for all $y\in H$. Show $||x_n-x|| \to 0$. My attempt: I'm ...
0
votes
2answers
36 views

If $X \subset \mathbb{R}^n$ is a set bounded then $\forall a\in R^n,$ $X\subset B[a,r]$.

I need to know whether the proposition is properly demonstrated. But first consider the following: Definition. A set $X \subset \mathbb{R}^n$ is bounded if and only if there exists $c\in \mathbb{R}$ ...
1
vote
1answer
52 views

Estimate the integral of a function on the boundary using the integral of its gradient

Show that for a sufficiently smooth boundary of $\omega$ and any $\epsilon > 0$, there exists $C$ such that $$\int_{\partial \omega} u^2\ ds \leq C\int_{\omega} u^2\ dx + \epsilon \int_{\omega} |\...
2
votes
1answer
49 views

When does $\sum_{n=0}^\infty \frac{a_n x_n}{n!}f(b_nx)$ converge for given $f\in C_c^\infty(\Bbb{R})$ and $(a_n)_{n=0}^\infty$?

Let $f\in C_c^\infty(\Bbb{R})$ be such that $f(x)=1$ for $x\in (-1,1)$. Given a real sequence $(a_n)$, define $$ g(x):=\sum_{n=0}^\infty \frac{a_n x^n}{n!}f(b_nx), \quad x\in\Bbb{R} $$ where $(...
1
vote
1answer
86 views

Riemann integration of a convex function

Let $f$ be integrable convex function on $[a,b]$. Prove $$f\left(\frac{a+b}{2}\right) \leq \frac{1}{b-a} \int_a^b f\, \mathrm{d}x.$$ Intuitively, I see that this means the value of the function at ...
1
vote
1answer
107 views

Can a smooth convex functions be non-differentiable?

Consider the definition of the $\beta$-smoothness (for some constant $\beta$): $$ \|\left. \nabla f \right|_{ y } - \left. \nabla f \right|_{ x } \| \leq \beta \| x - y \| $$ And convexity: $$ f(x)...
0
votes
1answer
78 views

Show a function is not Lipschitz Continuous [duplicate]

First I constructed the negation to Lipschitz continuity: $ \forall L > 0, \exists x, y \in [a,b]\ |f(x) - f(y)| > L|x - y| $ For $f(x) = \sqrt x$, notice $$|f(x) - f(y)| = |\sqrt x -\sqrt y| ...
4
votes
0answers
70 views

Suppose that $f:[a,b] \to \mathbb{R}$ is differentiable. Show that if $f(a)=f(b)=0$, then $|f'(c)|\ge \frac{4}{(b-a)^2}\int_a^b f(x)dx$. [duplicate]

I have tried to solve the below problem, and think I went very close to getting the answer but have been unsuccessful so far. Below is the problem and my proof. Suppose that $f:[a,b] \to \mathbb{R}$ ...
2
votes
1answer
68 views

If $f \in C([0,1])$, show $f\in C^{k}([0,1])$ iff $f$ is $k$ times continuously differentiable on $(0,1)$ and has derivative at endpoints

The following is from question 5.9 in Folland's Real Analysis: Modern Techniques and Their Applications. Let $C^{k}([0,1])$ be the space of functions on $[0,1]$ possessing continuous derivatives ...
0
votes
1answer
31 views

Does measure imply the existence of limit function?

I am reading a book and I don't quite understand some of the statements. It says "$\{u_n\}$ is a Cauchy sequence in the space $L^2(\mathbb{R}^d;|\xi|^{2s}d\xi)$. Because $|\xi|^{2s}d\xi$ is a measure ...
0
votes
1answer
30 views

Infinite Metric space, limit points.

Prove that in every infinite metric space, there exists an infinite sequence $(x_n)_1^{\infty}$ where no limit point of the set $S = \{x_1,x_2,\dots\}$ is an element of the sequence. Thoughts on how ...
0
votes
1answer
28 views

In an interval do the infinum, supremum, upperbound and lower bound needed to be in the interval

There are four different parameters for an interval. There are upper bound, lower bound, infinum and supremum. Out of the above four which of them must be within the interval, which may be within the ...
2
votes
1answer
34 views

Compute the norm of the operation $A$

Suppose that $\left ( a_{ij} \right )_{i,j=1}^{\infty}$ is a matrix satisfying the following condition $$\sum_{i,j=1}^{\infty} \left | a_{ij} \right |^q < \infty$$ where $q>1$. For $x=\left \{ \...
0
votes
2answers
29 views

Normalization in $L^{p}$ and $L^{q}$

Given a function $f$ in $L^{p}\cap L^{q}$ where $0<p,q<\infty$, can $f$ always be normalized such that $\| f \|_p=\| f \|_q=1$?
0
votes
1answer
45 views

Sequence of measurable functions $f_n$ on $\mathbb R^n$ converges to $0$ in measure

I am having problems on how to start, any hints are appreciated. Thank you. Show that a sequence of measurable functions $f_n$ on $\mathbb R^n$ converges to $0$ in measure if and only if $$ \...
1
vote
1answer
56 views

Property of nth root

I'm trying to prove the following result: "Let $x, y \geq 0$ be non-negative reals, and let $n, m \geq 1$ be positive integers. If $x > 1$, then $x^{1/k}$ is a decreasing function of $k$." $x^{...
4
votes
1answer
61 views

Uniformly continuous in the compact variable

What is this theorem/where can I find a proof of the following? Let $f: K \times X \to Y$ be continuous, with $K$ compact. Using uniform continuity ideas, for $x \in X$ and $\epsilon > 0$, ...
2
votes
1answer
26 views

Admits partials, definite integral of multivariable function is $C^1$.

For open $U \subseteq \mathbb{R}^n$, assume $f: [a, b] \times U \to \mathbb{R}$ admits partials $\partial_{x_i}f(t, x_1, \dots, x_n)$ continuous on $[a, b] \times U$. Do we have that $I_f(x_1, \dots, ...
8
votes
3answers
700 views

Equivalence of Rolle's theorem, the mean value theorem, and the least upper bound property?

How to show that Rolle's theorem, the Mean Value Theorem are equivalent to the least upper bound property? I'm thinking of starting like this: Let F be an ordered field that does not satisfy the ...
1
vote
0answers
53 views

Application of Radon-Nikodym

Let $\nu$ and $\mu$ be two $\sigma$-finite measures on $(X,\mathcal{M})$ and assume that RN holds, with $w = \frac{d\nu}{d\mu}$. Show that, for each non negative measurable function $g$ it holds $$ \...
4
votes
1answer
102 views

Confused by Monotone class theorem for functions

Monotone Class Theorem has two types. One is Monotone class theorem for sets and the other for functions. I have no doubt for sets. Here is a reference of definition of Monotone Class Theorem for ...
1
vote
1answer
64 views

Relation between absolute continuity with respect to lebesgue measure and compact sets on R

Let $\mu$ be a $\sigma$-finite measure on $(\mathbb{R},\text{Borel sets on }\mathbb{R})$, which is absolutely continuous with respect to the Lebesgue measure. Then, is it true that, $\mu(K) < \...
0
votes
1answer
90 views

Proof that a number and its multiplicative inverse have the same sign

I am trying to prove that if a number $a > 0$, then its multiplicative inverse $a^{-1}$ is also $>0$. What I have done thus far is this: Using the trichotomy property for the real numbers, I ...
3
votes
1answer
373 views

let $A_1, A_2, A_3, \dots$ be a collection of nonempty sets, each of which is bounded above. $(a)$ Find a formula for $\sup(A_1\cup A_2)$.

Let $A_1, A_2, A_3,\dots$ be a collection of nonempty sets, each of which is bounded above. $(a)$ Find a formula for $\sup(A_1 \cup A_2)$. Extend this to supremum of a collection of $n$ sets $A_1,...
1
vote
2answers
108 views

Find smallest and largest values in a bounded interval such that a function equals a value

I have the following problem: If $v$ is a value of a continuous function $f:[a,b] \rightarrow \mathbb{R}$, use the least upper bound property to prove that there are smallest and largest $x \in [a,b]$ ...