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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0answers
42 views

Is there a math basis for what I observe in simulations?

I am stuck in my research with the following problem which occurs when proving a theorem. If $x,y\in \mathbb{R}$, and $x>y$, is it true that $(x-y)^{((x-y)/(2x-y))}\times ...
1
vote
0answers
13 views

Prove boundedness of 2nd derivatives

Let $f \, \colon \mathbb{R}^d \rightarrow \mathbb{R}^d$ be a smooth and convex function. Assume $f$ behaves asymptotically as a cone at infinity, i.e., $ \lim_{R \rightarrow \infty} \frac{f(R x)}{R} ...
-1
votes
0answers
16 views

Prob.8, Sec. 3.8 in Erwine Kreyszig's INTORDUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: A Hilbert space is isomorphic to its second dual?

How to show that any Hilbert space $H$ is isomorphic to its second dual space $H^{\prime\prime} = (H^\prime)^\prime$? Given a real or complex normed space $X$, its dual space $X^\prime$ is defined ...
-3
votes
0answers
39 views

Proving using AM-GM inequality

If $x,y\in \mathbb{R}$, and $x>y$, how to show $(x-y)^{((x-y)/(2x-y))}\times (x+y)^{((x)/(2x-y))}>x$? I know I have to use AM-GM inequality, but it is not clear how.
0
votes
1answer
26 views

Proving that a solution exists

Proof that there exists a $x>0$ with $x \in \mathbb{R}$ s.t. $\sin(x) = \frac{x}{2}$ I tried to use the intermediate value theorem, but I don't know how to apply it correctly. Obviously ...
0
votes
1answer
16 views

Sum of elements in a sequence

Let $a_n$ be a sequence in $\mathbb{R}$ and $a\in\mathbb{R}$. Suppose that $N \in \mathbb{N}$, $\epsilon >0$ and for every $n > N$ $|a_n -a|<\epsilon$. Show that for every $n>N$ the ...
0
votes
3answers
63 views

Is $\sqrt{2t \log{\log{\frac{1}{t}}}}$ increasing for $t\in(0,a)$, for a suitable $a>0$?

I believe it's obviously, but I tried a lot, and have no clue how can I show that $$\sqrt{2t \log{\log{\frac{1}{t}}}}$$ is increasing for $t\in(0,a)$ for a suitable $a>0$?
0
votes
1answer
36 views

Directional derivative $f(x,y)=\frac{x^3}{1+x^2+y^2}$

I'm stuck on calculating the directional derivative of $f(x,y)=\frac{x^3}{1+x^2+y^2}$ in $(3,-1)$ along $(a,b)\in\mathbb{R}^2$. My try: $\lim\limits_{t\to ...
4
votes
0answers
30 views

$L^2$ convergence of this sequence

I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 ...
2
votes
1answer
26 views

Simple points of an algebraic variety from an analytic point of view

I am a specialist in fuctional analysis, but from time to time I have to use some results from algebraic geometry, and every time I face great difficulties in translating them into the language ...
2
votes
1answer
30 views

Is there a total summation function?

I define a summation function to be a partial function $F$ from infinite sequences of real numbers to the extended reals, such that: (1) Sequences which are zero in all but possibly one position are ...
-1
votes
0answers
27 views

How can I resolve: $ 2x'' - 5x' - 3x = 45e^{2t}, x(0)=2 \text{ and }x'(0)=1 $ via numerical solution?

How can I resolve a second-order ODE via Euler method? By example in the next ODE: $$ 2x'' - 5x' - 3x = 45e^{2t}, x(0)=2 \text{ and }x'(0)=1 $$ I know Euler method: $x_{i+1} = x_{i} + ...
1
vote
1answer
21 views

Showing an integral is in $L^1$

Let $0<a<1$ and $f\in L^1([0,1])$. Show $g(x)=\int_0 ^x\frac{1}{(x-t)^a}f(t)dt$ exists a.e. in $[0,1]$ and $g\in L^1([0,1])$. Using Fubini, $$\int_0 ^1 \vert g(x) \vert dx=\int_0 ^1 \int_0 ...
1
vote
1answer
28 views

Prob. 7, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: The dual space of a Hilbert space is a Hilbert space.

Here's Prob. 7, Sec. 3.8 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that the dual space $H^\prime$ of a Hilbert space $H$ is a Hilbert space with inner product ...
-1
votes
1answer
15 views

Continuous function and its Mapping

Let $f: \mathbb R \to \mathbb R$ be a continuous function. Which one of the following sets cannot be the image of $(0,1]$ under $f$? $\{0\}$ $(0,1)$ $[0,1)$ $[0,1]$. We know that $(0,1]$ is ...
3
votes
2answers
68 views

Prove this limit $\lim \limits_{x\to\infty}f(x)=0$

I have this problem in real analysis. I think it needs integral factor or knowledge of ODE to prove, but not sure how to it. Here is the question: Let $f$ be a real valued continuous function on ...
1
vote
1answer
14 views

Proving a corollary of a corollary of the Mean Value Theorem (corollary-ception)

This is will a wordy question but here it goes: My analysis book states the mean-value theorem and then a corollary which we will label as (1): Let $f$ be a differentiable function on $(a,b)$ such ...
-1
votes
0answers
12 views

$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n)$

I am trying to prove the equality $$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n),$$where $\mathcal B(\mathbb R^i)$ is the Borel $\sigma$-algebra on $\mathbb ...
3
votes
2answers
39 views

Question on proof of Heine-Borel theorem

Spivak's book on calculus on manifolds has a statement that I can't grasp. Say we have the closed interval $[a,b]\subset\mathbb{R}$ covered by $\mathcal{O}$ and we define $$A=\{x \in [a,b]:[a,x] ...
1
vote
0answers
14 views

inverse function theorem for analytic functions whose derivative might vanish

Suppose $x(t), y(t)$ are monotone increasing functions, and $f$ and $g$ are real-analytic functions that are not identically zero. If $f(x(t)) = g(y(t))$ for all $t$, does it follow that $x$ is an ...
0
votes
1answer
31 views

if $\lim a_n = \infty$ and $\lim b_n = B$, then $\lim (a_n+b_n) = \infty$

I'm having trouble starting the proof not sure exactly how to go about it. So far I know that for a sequence to go to infinity it means that for all $n >0$ there exists $n_0$ for all $n$ greater ...
3
votes
1answer
45 views

A function with midpoint-linear derivative is a quadratic polynomial

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $$f'\left(\frac{a+b}{2}\right) = \frac{f'(a)+f'(b)}2,\quad \forall a,b\in\mathbb{R}$$ Prove that $f$ is a polynomial of ...
0
votes
0answers
25 views

An interval covering problem [on hold]

Consider a set $A$ of intervals $[a_i,b_i)$, whose union is $[0,1)$. Prove there must exist a subset $B$ of $A$ such that the intervals in $B$ are pairwise non-overlapped and the sum of their lengths ...
1
vote
3answers
47 views

A recursive sequence is defined by…

A sequence is defined recursively by $a_1=1$ and $a_{n+1} = 1 + \frac{1}{1+a_{n}}$. Find the first eight terms of the sequence $a_n$. What do you notice about the odd terms and the even terms? By ...
0
votes
0answers
9 views

Comparison between interpolation and Tikhonov regularization.

Interpolation is defined as finding a value of a function between two points and one can think of Tikhonov regularization as to estimate a suitable function under certain condition. Can we think ...
0
votes
1answer
33 views

W.Rudin (1.22 decimals)

I am reading the book of Rudin "Principles of mathematical analysis". In 1.22 decimals he wrote: Let $x>0$be real. Let $n_0$ be the largest integer such that $n_0\leqslant x$ (Note that the ...
1
vote
3answers
57 views

$f(x)=\sum_{n=0}^{\infty}a_n x^n$ and there exists a sequence $(x_n)$ tending to $0$ such that $f(x_n)=0$ for all $n$, then $f(x)=0$ for all $x$.

I found this question really difficult for me, I don't even know how to start with it? Could you help me? I will appreciate that. Prove that if $f(x)=\sum_{n=0}^{\infty}a_n x^n$ (defined in ...
1
vote
1answer
20 views

Show a Function is strictly monotone Increasing, and what does it say about its inverse?

For example: $$g(x)=x^3-3x^2-1 \quad, \quad x\in [2,+\infty]$$ What I have tried to do was to take the first Derivative. I get $$ g'(x)=3x^2-6x$$ I then check the sign of Derivative of g(x) at ...
2
votes
2answers
43 views

If a $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how to find sequence $\{b_n\}$ such that $\sum |b_n|<\infty$ but $\sum |a_n||b_n|$ diverges?

If we are given any sequence of real numbers $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how can we find a sequence $\{b_n\}$ such that $\sum |b_n|$ converges but $\sum |a_n||b_n|$ diverges? I ...
1
vote
1answer
20 views

Existence of a smooth function with given derivative roots

Is there a smooth function $f$ that for all $n\in\mathbb{Z}_+$, $f^{(n)}(n)=0$ i.e. $n$th derivative at the point $n$ is zero and $f^{(n)}(x)\ne 0$ for all $x\in\mathbb R\setminus \{n\}$? If there is ...
0
votes
1answer
31 views

Which of two quantities is greater?

Let $x$ and $y$ be two positive real numbers such that $x>y$. Which of the quantities is bigger and when? $(x-y)\log\left(1-\frac{y}{x}\right)$ $x\log\left(1-\frac{y}{x+y}\right)$
2
votes
0answers
28 views

Suppose that $f:(0,\infty)\rightarrow \mathbb{R}$ is a function satisfying $f'(x)=1/x$ for all $x\in (0,\infty)$, show $f(xy)=f(x)+f(y)$. [duplicate]

I would like to ask you for some help with the following problem. Suppose that $f:(0,\infty)\rightarrow \mathbb{R}$ is a function satisfying $f'(x)=1/x$ for all $x\in (0,\infty)$, and $f(1)=0$. Show, ...
0
votes
1answer
27 views

Construction of a smooth function $\varphi$ such that the sum of $\varphi(2^{-n}t)$ is constant

I want to construct $\varphi \in C_c(\mathbb R)$ whose support is in $[\frac 1 2 ,2]$, satisfying $\displaystyle\sum_{n=-\infty}^\infty \varphi(2^{-n}t)=1$ for all $t>0$. I guess I should use ...
1
vote
1answer
17 views

Heat Equation on $[0,l]$ with Neumann boundary conditions

I was reading the following pdf about the heat equation on an interval $[0,l]$ with Neumann conditions, http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_N_web/heat_ex_homo_neum.pdf i.e. ...
1
vote
0answers
29 views

Understanding series and their sums

Here's something that I can't wrap my head around while self-studying analysis. Is defining a function to be a series and defining a function to be the sum of a series considered to be two different ...
3
votes
1answer
32 views

Extreme values of a two-variable polynomial

Is it possible to find a two-variable polynomial which has only two extreme values on the whole plane, one is a local maximum, another is a local minimum, and the local maximum is less than the local ...
-2
votes
0answers
30 views

The $\{ 0,1\}^{\mathbb{N}}$ bit-sequences for the metric space.

Let $M=\{ 0,1\}^{\mathbb{N}}$ denote the set of bit-sequences. For two bit-sequences $x=(x_{n})_{n\in \mathbb{N}}$ and $y=(y_{n})_{n\in \mathbb{N}}$ let $\mu(x,y)=\min\{ n\in \mathbb{N}\mid ...
0
votes
2answers
26 views

How to evaluate limits

Let $f$ be a continuously differentiable function on $\mathbb R$. Suppose that $L=\lim\limits_{x\to \infty}(f(x)+f^{'}(x))$ exists. If $0<L<\infty$, and if $\lim\limits_{x\to \infty} f^{'}(x)$ ...
0
votes
1answer
33 views

Spivak Ch1 Proof Critiques

I've started working through Spivak's Calculus. I'm going into senior year after this summer, took the AP Calculus BC test last year, and wanted to get a firmer foundation in calculus before I take ...
1
vote
4answers
48 views

General expression for the k-th derivative of $(\cos(x))^n$

Is there a general expression for the higher-order derivatives of $(\cos(x))^n$ evaluated at the origin? The odd derivatives are zero due to the symmetry, but what about the even derivatives?
0
votes
0answers
23 views

Showing sequence function is monotone [duplicate]

Is this sequence of functions monotone? $$f_{n+1}(x)=\varphi(x)f_n(x)+\frac{1}{\varphi(x) f_n(x)}, \forall x \in[0,+\infty)$$ Where $\varphi:[0,+\infty)\to \mathbb{R}$, $1/2\leq \varphi(x) <1$ ...
1
vote
0answers
18 views

Non-asymptotic error bound

I am looking for a sufficiently tight estimate of the following over the interval $[-T,T]$: $| \exp(t) - (1+t/n)^n|$. This is of course $o_n(1)$. What I am looking for is a non-asymptotic estimate ...
0
votes
1answer
30 views

Is point-to-set distance function $C^\infty$ for $\mathbb{R}^n$

Let $x\in \mathbb{R}^n$ and $Q\subset \mathbb{R}^n$. Then we define the point-to-set distance function as: $$ d_Q(x) = \inf_{y \in Q} \| x-y\| $$ It's continuous for every normal space (not only ...
1
vote
0answers
10 views

An optimization problem with a simplex constraint

Suppose $X^i=[0,1]$ for $i=1,2,3$. $X=\prod_i X^i$ and $\mu_i$ is a measure on $B([0,1])$ and $\mu$ is the product measure. Let $f,g,h$ be $L^2(\mu)$ integrable functions satisfying $$0\leq ...
4
votes
5answers
87 views

Why is $\operatorname{Int}(A) \cup \operatorname{Int}(B) \neq \operatorname{Int}(A \cup B)$?

I know that $\operatorname{Int}(A) \cup \operatorname{Int}(B) \subset \operatorname{Int}(A \cup B)$, but that the other direction does not hold, so can anybody please tell me whats wrong with the ...
0
votes
2answers
38 views

Is $t\mapsto 1_{[0,t]}(s)$ for a fixed $s\ge 0$ continous?

Let $s\ge 0$ and $$f:[0,\infty)\to\left\{0,1\right\}\;,\;\;\;t\mapsto 1_{[0,t]}(s)$$ Is $f$ continuous at $t_0\ge 0$? If $s>t_0$, then $f(t_0)=0=\displaystyle\lim_{n\to\infty}f(t_n)$ for all ...
1
vote
2answers
42 views

under which conditions this equality holds

Consider $f : [0,\infty) \rightarrow \mathbb{R}$ be a function such that $\lim_{t\rightarrow \infty} f(t) = 0$. I was wondering if the following relation holds $$lim_{t\rightarrow\infty}\int_0^t ...
0
votes
0answers
25 views

Normal coordinates

I was wondering if this is a legitimate way to define the induced basis of the tangent space in normal coordinates. So the exponential map is a diffemorphism $exp:U \subset T_pM \rightarrow V \subset ...
1
vote
0answers
28 views

What properties does $a(n)$ have to fullfill to get $\log(2^{cn}+a(n))\sim\log(2^{cn})$?

Let $c$ be an exponential growth rate, and $a(n)$ any expression in $n$ (sequence, polynom, function,...). Consider $$ \log(2^{cn}+a(n)). $$ I am asking myself what properties (increasing, ...
0
votes
0answers
11 views

Polar coordinates: rescale radius and maintain smoothness

I am looking for a transformation that takes a smooth function $f^\in C^\infty(\overline{B}_1(0))$ on the unit ball, say in $\mathbb{R}^2$, and makes it "even smoother" at the boundary. In 1D, ...