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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5
votes
2answers
123 views

Difficult limit question

Prove that for real number $p\geq 0$ we have $$\lim_{n\to\infty}\frac{(1^{1^p} \cdot2^{2^p}\cdots n^{n^p})^\frac{1}{n^{p+1}}}{n^\frac{1}{(p+1)}} = \exp\left(\dfrac{-1}{(p+1)^2}\right).$$
0
votes
1answer
72 views

Homework - Showing any continuous functions on a compact subset of $\mathbb{R}^3$ can be approximated by a polynomial.

$ X = \left\{(x, y, z) | \frac{x^2}{3} + \frac{y^2}{5} + \frac{z^2}{7} \le 1 \right\} $ Prove: If f(x,y,z) is continuous on X, then for any ϵ > 0, there exists a polynomial p(x,y,z) such that ...
3
votes
3answers
135 views

Show the limit of the following is $\dfrac{1}{12}$

Show that $$\lim_{n\to \infty}n^2 \log \left(\dfrac{r_{n+1}}{r_{n}} \right)=\dfrac{1}{12}$$ where $r_{n}$ is defined as; $$r_{n}=\dfrac{\sqrt{n}}{n!} \left(\dfrac{n}{e} \right)^n$$. Now I ...
1
vote
1answer
95 views

Elliptic PDE - max principle

The maximum principle for elliptic PDEs is established for the nondivergence form as in http://www.ann.jussieu.fr/~frey/cours/UdC/ma691/ma691_ch3.pdf. But what if we are dealing with the divergence ...
0
votes
0answers
26 views

Constraint approximation in non-linear optimization

In given non-linear optimization problem \begin{equation*} \begin{aligned} & \underset{x \in\mathbb R^n}{\text{maximize}} & & f(x) = \alpha^2 \\ & \text{subject to} & c(p(x)) \le ...
1
vote
1answer
62 views

Does equality of ranges of $fh$ and $gh$ for all $h$ continuous imply equality of $f$ and $g$?

Let $X$ be a compact Hausdorff space and suppose Range($fh$)=Range($gh$) $\forall h\in C(X), f, g$ also in $C(X)$. The question is whether this implies that $f=g$. Let us for the time being assume ...
8
votes
4answers
399 views

Integral $\int_0^{\pi/2} \frac{\sin^3 x\log \sin x}{\sqrt{1+\sin^2 x}}dx=\frac{\ln 2 -1}{4}$

Hi I am trying to prove$$ I:=\int_0^{\pi/2} \frac{\sin^3 x\log \sin x}{\sqrt{1+\sin^2 x}}dx=\frac{\ln 2 -1}{4}. $$ Thanks. I am possibly trying to simplify this to obtain something like ...
0
votes
1answer
32 views

If lim $s_n = + \infty$ and inf$\{t_n | n \in \mathbb{N}\} > - \infty$, then lim $(s_n + t_n) = + \infty$

Here's the question: If lim $s_n = + \infty$ and inf$\{t_n | n \in \mathbb{N}\} > - \infty$, then prove that lim $(s_n + t_n) = + \infty$ Here's my proof. Can someone please verify it or offer ...
-2
votes
1answer
67 views

Closed and Compactness on $\mathbb Q$ (Multiple Choice)

Please help me regarding the following question. Consider $\mathbb Q$ with usual metric (i.e $d(p,q)=|p-q|$).Then which of the following are true? $\{q\in\mathbb Q|2<q^2<3\}$ is closed ...
1
vote
1answer
59 views

Integrate over different measures

In Probability theory the expected value of a random variables $X : \Omega \rightarrow \mathbb{R}$ is defined as $E(X) = \int_\Omega X dP$ Now, if $\Omega \subset \mathbb{R}$ and has a density ...
-1
votes
2answers
86 views

Discontinuities of $f(x)=\begin{cases}|x^2-1| & \text{if } x \text{ is irrational} \\0 & \text{if } x \text{ is rational}\end{cases}$

$f(x)=\begin{cases}|x^2-1| & \text{if } x \text{ is irrational} \\0 & \text{if } x \text{ is rational}\end{cases}$ what can we say about the number of points of continuity of the function ...
2
votes
2answers
95 views

Discontinuous functions with the intermediate value property

Show that any function $f$ which is not continuous on $[a,b]$, but satisfies the intermediate value property, assumes some value infinitely often. Here $f$ has the intermediate value property if: ...
3
votes
2answers
60 views

Two sequences are equivalent. Prove that one is Cauchy iff the other is Cauchy.

This question has already been asked and answered here Let $ϵ>0$ be given. With loss of generality, we may assume $ϵ$ is rational. Suppose $a_n$ is a Cauchy sequence and $b_n, a_n$ are ...
0
votes
2answers
49 views

Analysis Limit Question Confusion

Suppose that $|r| < 1$. Show that $$1 + r + r^2 + \cdots + r^n = \frac{1 - r^{n+1}}{1 - r} $$ and find $\lim_{n \to \infty} (1 + r + r^2 + \cdots + r^n)$ Does anyone have any idea on how to ...
0
votes
2answers
75 views

Proving the completeness of $\mathcal{L}(\mathcal{H})$

Here $\mathcal{L}(\mathcal{H})$ denotes the vector space of all bounded linear operators on a Hilbert space $\mathcal{H}$. We can define a norm on $\mathcal{L}(\mathcal{H})$ by $\|T\| = \inf\{B : ...
1
vote
1answer
87 views

double integral.

I just received this problems from a friend, and I think its a HW problem. its: $$ \int_1^e \int_{1+y^2}^5 \cos (x- \ln x) \ dx \ dy $$ I looked at it, and If I did graph the the region right then ...
4
votes
2answers
172 views

Compact metric connected space

If I have a compact metric space $X$ such that for all $a,b \in X$, there are points $a:=x_1,...x_n=:b$ such that $d(x_i,x_{i+1})< \varepsilon$, then this space is connected. Somehow, I don't see ...
0
votes
0answers
22 views

Conditions on convergence of a series [duplicate]

Suppose we have a sequence $(a_n) \to 0$ and the partial sums $\sum_{k=1}^{n} a_k$ are uniformly bounded. Does the series converge? I think the answer should be no, since, for example, the Dirichlet ...
1
vote
1answer
58 views

Measure theory $L^p$ and $L^q$ spaces

For any two different numbers $p,q\in[1,\infty)$ find functions $f\in L^p \setminus L^q$ and $g\in L^q \setminus L^p$. Idea: This and the function $f=x^{-1/p}(1+|log x|)^{-2/p}$ and then I need to ...
1
vote
1answer
54 views

Conclude that if all but finitely many $s_n$ belong to $[a, b]$, lim $s_n$ belongs to $[a,b]$

Here's the question: Let $(s_n)$ be a convergent sequence. Conclude that if all but finitely many $s_n$ belong to $[a, b]$, lim $s_n$ belongs to $[a,b]$ Here's my attempt at a proof. Can someone ...
3
votes
0answers
73 views

How come Stone-Weierstrass theorem does not imply that in a given interval every continuous function has a power series expansion?

Since for all continuous functions we get a polynomial sequence that uniformly converges to that function? As the degree of polynomial increases it should look like a power series expansion?
2
votes
2answers
61 views

Some confusion regrading $\sigma$-algebras

I have the following proposition: Let $X$ be a non-empty set and $\mathcal{B}$ a collection of subsets of $X$. Then there exists a smallest $\sigma$-algebra $\sigma(\mathcal{B})$ containing ...
1
vote
1answer
67 views

Defining the number $ e $

In this YouTube video it is said that $ e $ naturally arises as a number that allows us to take the derivatives of functions like $ a^x $. So $ e $ is defined as a number for which: ...
1
vote
1answer
68 views

$\frac{1}{(1+s^{2}) (1+t^{2})}$ real analytic in $\mathbb R^{2}$ but not real-entire; why?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
2
votes
1answer
47 views

Proving there exists $ r$ so that the limit is $1 $

How would I go about proving there exists a real number $ r $ such that: $$\lim_{x\rightarrow 0}\frac {r^x-1}{x}=1$$ and actually finding its value?
0
votes
0answers
161 views

Proving a Sequence is Contractive

I'm trying to prove that for $a_1 \neq a_2$, with $a_1, a_2 \in \Re$, $a < b < 1$; and $n \geq 3$, where $a_n = ba_{n-1} + (1-b)a_{n-2}$, that the sequence {$a_n$} is contractive. I've looked ...
1
vote
0answers
30 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
14
votes
1answer
413 views

Integral $\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
6
votes
1answer
419 views

Is this a proper proof of (-1)(-1) = 1?

I am a novice in proof writing and have just started a book on analysis. I have no other pure math experience or knowledge of abstract algebra. I am trying to prove that $(-1)*(-1) = 1$. I will ...
3
votes
0answers
78 views

Law of iterated logarithms for BM

The law of iterated logarithms for the standard Brownian motion asserts that $(\ast) \limsup\limits_{h \downarrow 0} \frac{B(h)}{\sqrt{2h\log\log(\frac{1}{h})}} = 1$ I'm trying to prove the ...
0
votes
1answer
5 views

How can you tell that a multivariate nonlinear integral function is parabolic?

The article *“Neural-Gas” Network for Vector Quantization and its Application to Time-Series Prediction*$^1$ presents the function \begin{equation} E_\lambda = \frac{1}{2 C(\lambda)} \sum_{i=1}^N ...
1
vote
2answers
90 views

How to prove that a function is continuous?

Could you give me some hint how to solve this question: Suppose $f$ is a differentiable function for all $0<x<1$,$f(0)=1,f'(x)>0$ in the given interval. It is obvious that $f$ is continuous ...
1
vote
2answers
54 views

How can I show that ~ is an equivalence relation such that $x$~$y$ if there is a continuous path in $M$ from $x$ to $y$?

Given a metric space $M$, we will define a relation ~ on $M$ by defining $x$~$y$ if there is a continuous path in $M$ from $x$ to $y$. I'd like to show that ~ is an equivalence relation. I am ...
0
votes
1answer
151 views

Showing that the norm of the composition of linear operators is bounded by the product of the norms of the individual linear operators [closed]

I would like to use the definition of the linear operator norm to show that $\|S\circ{}T\| \leq \|S\|\cdot\|T\|$ and I am not sure how to make progress on this problem.
0
votes
1answer
26 views

Basic question about set theory and dilations

Suppose we have an open ball $B_1 = B(0, \frac{1}{2} ) \subset X$ where $X$ is complete normed space. $B_1$ is open ball centered at $0$ of radius $\frac{1}{2}$. Does it follow that for a given $x \in ...
0
votes
0answers
140 views

How can I envision the open ball around the french railways metric?

I have that the French railways metric defined by a metric space $(\mathbb{R}^2,d)$ has a distance function that is a metric as follows: $$d(x,y) = \begin{cases} \|x-y\|, & \text{if $x,y,0$ are ...
0
votes
1answer
31 views

A question about Maclaurin polynomial

Could you please give me some hint how to find 3-th degree Maclaurin polynomial of f(x) given f(0)=1 and for all $0<x<\lambda$ $f'(x)=1+f(x)^{10}$. If $\lim_{x\to0}f(x)=f(0)=1$ then $\lim_{x\to ...
0
votes
3answers
33 views

basic question about open balls

Let $U$ be an open set in a complete normed space. Let $x \in U$. Hence, we can find an open ball $B$ centered at $x$ that lies inside $U$. Question: Does it follow that $U - x $ contains an open ...
3
votes
0answers
40 views

Positive function approximation

Let $(X, \mathscr{S}, \mu)$ and $(Y, \mathscr{T}, \lambda)$ be two $\sigma$-finite measure spaces, give $X\times Y$ the product measure, then is it true that for any positive measurable function ...
0
votes
1answer
21 views

Convergence of a sequence of powers of functions

Let $f_n : [0,1] \to \mathbb{R}$, $n = 1,2,...$ be functions that converge uniformly to a function $f$, which is bounded. I wish to show that $f_n(t)^m$ converge uniformly on $[0,1]$ to $f(t)^m$ for ...
1
vote
2answers
63 views

Why is $L^p$ isomorphic to $(L^p)^2$

Is it possible to say why the spaces in the title are isomorphic as Banach spaces? Is their a Theorem that says this or is it even possible to find an explicit representation of this isomorphism?
0
votes
1answer
48 views

Converge of Sum divide by log(n)

I am trying to show that If $b_n = \sum^n_{k=1}(k^{-1}) -\sum^n_{k=1}(k^{-2})$ then $\frac{b_n}{\log(n)} \rightarrow 1$ as $n \rightarrow \infty$ I start this problem by showing this inequality ...
0
votes
2answers
52 views

Two quartics vs one single higher-order polynomial

I was reading an article called Computing the Cube Root, in which they approximate a cube root using one quartic polynomial divided by another, in the form ...
0
votes
1answer
53 views

Prove that lim $\sqrt{n^2+1}-n = 0$

Here's the question: Prove that lim $\sqrt{n^2+1}-n = 0$ Here's my attempt at a proof. Can someone please verify it or provide suggestions for improvement? Let $\epsilon > 0$. Pick $N \in ...
0
votes
3answers
31 views

If $A_1\cap…\cap A_n \neq \emptyset$, does $(A_1\cap…\cap A_n)^{c} =A_1^{c} \cup … \cup A_n^{c} = \emptyset$?

If I have some collection of sets such that $A_1\cap...\cap A_n \neq \emptyset$, then what happens if I apply the complement (denoted by superscript c) to both sides? i.e., $(A_1\cap...\cap A_n)^{c} ...
4
votes
1answer
54 views

Does the integral $\int_0^1\frac{\sin x}{\sqrt{x^3}}\cos\left(\frac1x\right)dx$ converge?

I tried to prove convergence this way: Suppose: $f(x)=\left|\frac{\sin x\cos\left(\frac1x\right)}{\sqrt{x^3}}\right|$, $g(x)=\left|\frac{\cos\frac1x}{\sqrt x}\right|$. $\lim_{x\to ...
0
votes
2answers
67 views

Prove that the two integrals are equal with periodic function

Problem: Let f be a real-valued continuous periodic function with period T. Prove that $$\int_{a}^{a+T}f(x)dx = \int_{0}^{T}f(x)dx$$ Any good hints? My following strategy is to try to show that the ...
3
votes
1answer
108 views

Proof that if $\lim s_n=0$ then $\lim\sqrt{s_n}=0$

Here's the question: Suppose $(s_n)$ is a sequence of non-negative real numbers, and $\lim( s_n) = 0$. Prove that $\lim(\sqrt{s_n})=0$. Here's my proof. Can someone please verify it or offer ...
1
vote
0answers
27 views

Sequences proof

Here's the question: $(s_n)$ and $(t_n)$ are sequences such that $|s_n| \leq t_n \ \forall n \in \mathbb{N}$, and lim $t_n = 0$. Prove that lim $s_n = 0$ Here's my attempt at a proof. Can someone ...
1
vote
1answer
26 views

uniform continuity of maximum of slice function

Let $f$ be a function from $[0,1] \times [0,1]$ to $\mathbb{R}$. Define $g : [0,1] \to \mathbb{R}$ as $g(x) = \max\{f(x,y) | y \in [0,1]\}$. Is this statement true? If $f$ is continuous on ...