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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-9
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0answers
35 views

Real analysis question am noob [on hold]

(a) Use the method of characteristic roots to solve an+2 − 5an+1 + 6an = 7n, n ≥ 0 where a0 = a1 = 1. [ 7 marks ] (b) Use the substitution method to ...
1
vote
2answers
36 views

a theory of transcendental functions?

Lately I've been interested in transcendental functions but as I tried to search for books or articles on the theory of transcendental functions, I only obtained irrelevant results (like calculus ...
0
votes
4answers
67 views

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$. Should I look at this series as: $\sum_{n=1}^{\infty}({n!x^{(n-1)!})x^{n}}$? I am really confues here. In addition, any attempt to ...
0
votes
0answers
81 views

How do i remove my Guilt? [on hold]

When i see a theorem ,which i cannot prove in real analysis ,i think about it but still i couldn't figure it out .Then i look for its solution ,after understanding the proof i feel very guilt that i ...
18
votes
3answers
939 views

Does there exist a continuous function from [0,1] to R that has uncountably many local maxima?

Does there exist a continuous function from $[0,1]$ to $R$ that has uncountably many strict local maxima?
0
votes
0answers
17 views

simple performance calculation dependent on different factors

I have to decide performance of some exam, Performance is dependant on few factors: factor1 : x1 out of y1 where x1<=y1, e.g. I checked 6 options out of 10, nearer the value of x1 to y1, better ...
-4
votes
0answers
24 views

This is a question from Real Analysis by NL Carothers [duplicate]

Prove that $\mathbb{N}$ contains uncountably many infinite subsets $(\mathbb{N}_a)_{a \in \mathbb{R}}$ such that $\mathbb{N}_{a} \cap \mathbb{N}_{b}$ is finite if $a \neq b$.
5
votes
4answers
222 views

Is it possible to work out the derivative of $e^x$ using the summation definition of $e = \sum_n 1/n!$?

So I know this question is a bit obtuse because usually we define $e$ in terms of the $\lim_{n \to \infty} (1 + 1/n)^n$ definition, and then compute derivatives of $e^x$ from there appealing to the ...
1
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2answers
23 views

Express g's Fourier coefficients using f's ones, if $g(x)=f(x+c)$.

The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$ I am asked to express g's coefficients as a combination of f's ones, given ...
5
votes
1answer
38 views

Can anyone prove D'Alembert Criterion (Dalambert) criterion for converging positive sequences?

This will most likely be on the exam, but it is not given in the text book. In my notebook I have this proof which I will type out, but it makes no sense. Here it goes: $$\text{D'Alembert ...
3
votes
1answer
25 views

Existence of a vector field which dominates the first local vector fields given by the charts of a locally finite covering

Let $M$ be a smooth manifold, let $\{U_i,\psi_i\}_{i\in I}$ be locally finite family of charts and let $K_i\subseteq U_i$ be compact subsets. Does there exist a vector field $X$ on $M$, such that ...
1
vote
1answer
32 views

Prove the concavity of the transformation from a concave function to another

Let's say we have $f_1$ and $f_2$, both strictly increasing and strictly concave on $[0,+\infty)$. $f_1(0)=f_2(0)=0$ and the difference $f_1-f_2$ is strictly positive and strictly increasing. That is, ...
8
votes
3answers
125 views

difficult problem in riemman integrals

Could anyone help me with the following problem? Because i have stuck. problem Let $f:[a,b]\rightarrow [0,\infty)$ be continuous and not the zero function. Prove that $$\lim_{n\to \infty} ...
2
votes
0answers
38 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
1
vote
1answer
41 views

Prove that the reciprocal of a polynomial function $f(x)$ is uniformly continuous on $R$.

Prove that the reciprocal of a polynomial function $f(x)$ is uniformly continuous on $R$. (It is provided that the reciprocal of the function exists. In other words, $f(x)$ is never zero for any ...
2
votes
1answer
42 views

Why is '1' the multiplicative identity of complex numbers and quaternions?

I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me ...
0
votes
0answers
25 views

Asymptotic approximation inside of an integral

Suppose we want to find $$ \lim_{n \rightarrow \infty} \int_{a}^b f_n(x) dx $$ for $a, b \in \mathbb{R}$, $a \le b$. Suppose there is a sequence of functions $g_n(x)$ such that $g_n(x) \sim f_n(x)$ ...
1
vote
5answers
52 views

Show that this limit is related to Euler number

I am calculating the limit $\lim_{n \rightarrow \infty} \left( \frac{n!^{\frac{1}{n}}}{n} \right)= \frac{1}{e}.$ I got this limit from wolframalpha, but don't know how to show this.wolframalpha
0
votes
1answer
26 views

Analysis Doubt on sequence and series of functions

I have seen in Rudin the following "if a compact class of bounded continuous functions on a compact metric space is not equi-continuous then that class contains a sequence which has no equi-continuous ...
4
votes
2answers
37 views

Mean value theorem for integration in two dimensions

The mean value theorem for integration says that, if $G$ is a continuous real-valued function defined over an interval, $G: [a,b] \to \mathbb{R}$, then the mean value of G on the interval is achieved ...
0
votes
1answer
34 views

There are two periodic functions $f(x)$ and $g(x)$, provide an example when $f(x)*g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions

There are two periodic functions $f(x)$ and $g(x)$ which are defined on $\mathbb{R}$, provide an example when $f(x)\cdot g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions ?
2
votes
1answer
37 views

Real analysis : Preliminary topics for - Measure Theory, Integration Theory, Differentiation and Integration [on hold]

I have following syllabus to study in Real Analysis Subject. I want to know, What are necessary topics that I have to cover as a prerequisite for below syllabus. Actually I am unable to get direction ...
0
votes
1answer
19 views

Trying to show a set equality regarding sigma algebras

Let $X_{\alpha}$, $\alpha \in A $ be a collection. Let $ X = \prod_{\alpha} X_{\alpha} $. Suppose $\mathcal{M}_{\alpha} $ is $\sigma-$algebra on $X_{\alpha} $. The product sigma algebra on $X$ is ...
2
votes
3answers
37 views

Limit in the set of real sequences.

I have troubles trying to prove the following proposition: Let $S$ be the set of real sequences with ...
0
votes
0answers
49 views

Multiple integrals of dirac delta

I'm working on a problem where integrals of this form arise: $$ \int\limits_{x_1=-1}^1 \;\; \int\limits_{x_2 =-1/3} ^{1/3} \dots \int\limits_{x_n =-1/(2n+1)} ^{1/(2n+1)} ...
2
votes
2answers
40 views

limit superior of sequences in real line

Let $(a_n)$ be a bounded sequence in $\mathbb R$. Is the limit superior of $(a_{n_k})$ less than or equal to the limit superior of $(a_n)$ for any subsequence $(a_{n_k})$ of $(a_n)$?
0
votes
1answer
54 views

When the function is continuous, bounded of variations, absolutely continuous?

Let the function $f_a:[0,1] \to \Bbb R$ be defined by $$f_a(x)=\begin{cases} x^a \cdot \cos(\frac{1}{x}) & 0 < x \leq 1 ;\\ 0 & x=0.\end{cases}$$ Find all values $a\ge 0$ such that ...
1
vote
1answer
35 views

Integral of a nonnegative Lebesgue-measurable function on $ [0,1] $.

Let $ f $ be a nonnegative Lebesgue-measurable function on $ [0,1] $. Suppose that $ f $ is bounded above by $ 1 $ and that $ \displaystyle \int_{[0,1]} f = 1 $. Problem. Show that $ f(x) = 1 $ ...
2
votes
4answers
45 views

$f$ is integrable & continuous over $[a,b]$ , $\int_{a}^{b}f(x)dx \geq 0$ for any subinterval $(\alpha,\beta)$ of $(a,b)$, then $f \geq 0$ in $[a,b]$

Some known things about this problem are: if $f(c) < 0$, $a < c < b$, then $f(x) < f(c)/2$ in some neighborhood of $c$, but I am not exactly sure how to use this to get to my goal of ...
1
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2answers
41 views

If $f$ is integrable on $[a,\ b]$ and $\int_a^b f(x) \mathrm dx >1$, then there exists a point $c$ in $(a,\ b)$ such that $f(c) > \frac{1}{b-a}$

So far for this problem, to my understanding, for something to be integrable means that $U(p,\ f) - L(p,\ f) < \epsilon$ but not sure how exactly to move beyond there to show that there exists a ...
0
votes
1answer
33 views

Looking for Clarification on a proof of Density of Q in R

I am looking for some advice/help in regard to the proof that Q is dense in R, given in Walter Rudin's book "Principles of Mathematical Analysis". Mostly, I want to see if my reasoning is correct for ...
-3
votes
0answers
48 views

Show smoothness of this map

Let $S^3$ be the sphere identified with the subset $\mathbb{C}^2$ as $\{(x,y) \in \mathbb{C}^2; |x|^2+|y|^2=1\}.$ Then I want to show that the map $\phi: S^3 \rightarrow \hat{\mathbb{C}}$ is ...
0
votes
0answers
16 views

How can i explain that: complete subspaces of R are separable implies that complete, unbounded subspaces of R contain unbounded sequences.

I know if a subspace of R is separable, thus exists a subset in that subspace that is enumerable dense, and what? I don't understand
0
votes
0answers
42 views

How to determine Taylor series expansion of function $f(x) = \frac{\cos(x)}{x}$ about $a=1$?

Given function is $f(x) = \frac{\cos(x)}{x}.$ $y = x - a , y = x - 1$. $x = y+1 , f(y) = \frac{\cos(y+1)}{y+1}$ How to get Taylor series expansion about $1$ of this function? If it was needed to ...
1
vote
0answers
21 views

If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...
0
votes
1answer
47 views

compute $\nabla f$ for a function over a cone

Let $D$ be the cone $D=\{rt:r>0, t\in\Omega\}$ with $\Omega\subset S^{n-1}$. I want to show that $$ \frac1{r^2}\int_{B_r}\frac{|\nabla f(x)|^2}{|x|^{n-2}} dx= C(n,g)r^{2(a-1)} $$ where $C(n,g)$ is ...
0
votes
2answers
58 views

dominated convergence theorem application

Prove or disprove this statement: if $f_n : \mathbb R \to \mathbb R$ are integrable functions with $f_n \to 0$ pointwise and $|f_n(x)| \le \frac{1}{|x| + 1}$ for all $n$, $x$, then $\lim\limits_{n ...
0
votes
0answers
12 views

What conditions are needed for cpt support of anti-gradient?

I'm reading through a paper by Michael Christ, "On the $\bar{\partial}_b$ Equation for Three-Dimensional CR Manifolds" found in the Proceedings of Symposia in Pure Mathematics, Volume 52, Part 3. One ...
2
votes
3answers
44 views

Prove that set is bounded but has no max/min

Okay so I'm reading through a proof in my text book and I dont understand one of the steps. We are proving that the set $(0,2)$ is bounded and has neither a max or min. We start by knowing that $0$ ...
3
votes
1answer
45 views

Proof $x_n \to \inf (A)$

$A$ is a non empty set of $\mathbb{R}$ with $\inf (A) = m$ Proof there exist a sequence $a_n$ with $a_n \to m$ [edit: is there an x = inf(A) then we can chosse for all n x_n := inf(A) ] * We ...
1
vote
3answers
94 views

Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$ [duplicate]

Determine the Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$. $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}} = \arcsin(x)\frac{1}{\sqrt{1-x^2}}$ It is known: (1.) ...
0
votes
0answers
27 views

Minimal boundary conditions for divergence theorem

I've noticed that some domain conditions of questions here were only supposed to be finite dimensional and bounded. And then the divergence theorem was applied in the answers. But if I'm not mistaken, ...
1
vote
1answer
17 views

An affine set $C$ contains every affine combinations of its points

Show that an affine set $C$ contains every affine combinations of its points. Proof by induction: From the definition of an affine set, we know that $\forall x_1,x_2\in C \text{ and } \theta_i\in ...
0
votes
2answers
41 views

How to find the Tangent space of a submanifold?

How can I determine the tangent space $T_pM$, $p\in M$, for the following submanifold? $$ M:= \{(x,y) \in \mathbb{R}^2|x^2-y=0\}\subset \mathbb{R},\quad p=(1,1) $$ I figured out, that I can ...
0
votes
1answer
19 views

How are the following inequalities concluded based on this first one?

$$I-\frac{\epsilon}{3} \leq s(f,T) \leq \underline{I} \leq \overline{I}\leq S(f,T) \leq I+ \frac{\epsilon}{3}$$ from this, the following is concluded, but how? $$1.\ \ \ 0 \leq |I-\underline{I}|\leq ...
4
votes
1answer
67 views

Group action and smooth manifolds

I was wondering if it is for a compact (i.e. Hausdorff) smooth manifold $M$ sufficient to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
0
votes
0answers
45 views

Limit tending to infinity

Given that $f(x)$ is continuous on $[0,\infty]$. If $\lim\limits_{x\to\infty}\left(f(x)+\int_{0}^xf(t)dt\right)$ exists then evaluate $\lim\limits_{x\to\infty}f(x)$
0
votes
1answer
35 views

How to find the image of an arbitrary element under this operator?

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T \colon H \to H$ such that $T e_n = e_{n+1}$ for $n = 1, 2, ...
0
votes
3answers
48 views

How derivatives of integrals like $G(x)=\int_{x-\sin x}^{\sin x}\arcsin(t)dt$ are computed?

I need to find the derivative of $G(x)=\int_{x-\sin x}^{\sin x}\arcsin(t)dt$. I know I don't have to find the integral, but I just have troubles computing the derivative. I suppose I get: ...
2
votes
2answers
76 views

If $|u+v| = |u| + |v|$ then $u = \lambda v$. How do I prove $\lambda \ge 0$?

I'm trying to prove that if $|u+v| = |u| + |v|$ implies $u = \lambda v$ for $\lambda \ge 0$. To this end I have $|u + v|^2 = (|u| + |v|)^2 \Rightarrow |u|^2 + |v|^2 + 2|u||v| = |u|^2 + |v|^2 + ...