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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1answer
16 views

Help on proof of claim about $limsup$ of real sequence.

Let $\{p_n\}$ be a sequence of real numbers. I would like to show that $~lim_{n \rightarrow \infty}sup~p_n< \infty$ if and only if $\{p_n\}$ is bounded above. Let ...
0
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0answers
15 views

$ l^{\infty}$ is not compact.

Prove that $l^{\infty}$ is not compact. My idea is: Let $ X=l^{\infty}$ defined by $ \{e^n=(0 ,0 .....\underset{nth}{1},0.......)\}$ This set is bounded and closed both but not totally bounded.(has ...
0
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1answer
67 views

Does $1-1+1/2-1/2+1/3-1/3+\cdots$ converges?

Is this true the the above series converge? What might be a economic way to show this? I tried to rearrange them but it seems like they are not all positive so its a bit dangerous. So it is ...
0
votes
1answer
9 views

Real 2D Analysis Question using Brouwer's Fixed Point Theorem

My question is as above. Currently I am stuck at the very start, part (i)! I can't come up with an appropriate $f$, even though I've been thinking about it for ages! If someone would be able to ...
1
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0answers
18 views

Proof check of sum of a compact and closed set of real numbers is closed

Let $A$ be a closed and $B$ be a closed and bounded set in $\mathbb R$ , then we have to show that $A+B:=\{a+b:a\in A , b\in B \}$ is closed in $\mathbb R$ . My Proof : Let $\{a_n+b_n\}$ be a ...
2
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2answers
31 views

Does $\sum\limits_{k=1}^{\infty} (a_{k+1} - a_k)^2 <\infty$ imply $\frac{1}{n^2} \sum\limits_{k=1}^n a_k^2 \to 0$?

I'm currently working on some problem regarding Dirichlet forms and, as the title states, I'm trying to figure out if $$\sum\limits_{k=1}^{\infty} (a_{k+1} - a_k)^2 <\infty \Rightarrow ...
0
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1answer
39 views

Is $\mathbb{R}^n$ not nowhere dense?

Is $\mathbb{R}^n$ not nowhere dense? I am trying to show that a clopen set is not necessarily nowhere dense. I know that it is both open and closed, but I am not sure how to find the interior of ...
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0answers
23 views

Taylor Series Expansion for Function of Two Variables (with Countable Discontinuities)

Given a real-valued function of two real variables, under certain conditions of smoothness in a closed ball about some point, we can obtain a Taylor series for the function about that point. I want ...
0
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1answer
32 views

Measurability of $\{(x,y): x\in M,0\leq y\leq f(x)\}$

Let $(X,\mathfrak{S}_x,\mu_x)$ be a measure space endowed with the $\sigma$-additive and complete measure $\mu_x$ defined on the $\sigma$-algebra $\mathfrak{S}_x$, let $\mu_y$ be the linear Lebesgue ...
0
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3answers
31 views

Proving one limit is equal to another

Let ($x_s$) be a convergent sequence, where $x_s>0$ for all s, and $y_s$ be a sequence such that $$\displaystyle \Large y_s=\frac{s}{\frac{1}{x_1}+...+\frac{1}{x_s}}$$ Prove $\displaystyle \lim ...
0
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2answers
38 views

What is $\int_0^{\infty} x^2e^{\frac{(x-\mu)^2}{2 a^2}} dx$?

How can we express the integral $\int_0^{\infty} x^2e^{-\frac{(x-\mu)^2}{2 a^2}} dx$ for example by means of the error function? The problem is of course, that the expectation value is shifted and we ...
0
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2answers
43 views

Is this sequence decreasing?

If a sequence $b_n>0$ and $b_n$ converges to $0$, can we say it is eventually decreasing? This problem bumps up when I am trying to something bigger. However, I am very unsure of this. If this is ...
0
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0answers
15 views

curve of class C1

Let $\mathbf{r}:(a,b)\to\mathbb{R}^2,\ r(t)=(x(t),y(t)), x,y\in C^{1}((a,b))$ be a non-constant function on any interval contained in $(a,b)$. Is it true that, if the following two limits exist: ...
10
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2answers
93 views

Test for convergence $\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$

What is the easiest way to test the convergence of $$\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$$ Is it possible to only use the high school tools for that?
0
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1answer
21 views

If $y$ is the solution of $\left\{y'=-y+\sqrt{t},y(0)=y_0>0\right\}$, then $\lim_{t\to\infty}\frac{y(t)}{\sqrt{t}}=1$

The homogeneous equation $$y'=-y$$ has the solution $$y_h(t)=ce^{-t}\;\;\;\;\;(c,t\in\mathbb{R})$$ In order to find a particular solution we can take the approach $$y_p(t)\stackrel{!}{=}c(t)e^{-t}$$ ...
1
vote
1answer
31 views

To show $(x+y)^p\leq x^p+y^p$, where $0\leq p\leq1, x>0,y>0$?

How to show that, $(x+y)^p\leq x^p+y^p$, where for $0\leq p\leq 1,x\geq 0, y\geq0?$ Any suggestion how to prove it? Thanks in advance.
0
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0answers
10 views

Area under mapping

Can some one help me with this problem: Let $f:R^2\rightarrow R^2$ be defined by $\displaystyle{f(x,y)=(e^{x+y},e^{x-y})}$ Find the area of the image of the region $\{(x,y) \in R^2 : 0<x,y<1 ...
1
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2answers
28 views

Convergence of cosine

Does the following sequence $$(\cos (\pi \sqrt{ n^2 + n}))_{n=1}^\infty $$ converge? I was trying the ratio or root test, but they don't seem to work in this case. Mean value theorem?!
12
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4answers
101 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
0
votes
0answers
25 views

Particular case of every sequence has a Cauchy subsequence?

A metric space (X,d) has the following property: Given $\epsilon >0$ and non-empty finite subset $X_\epsilon \subset X$ $$ \inf \{ d(x,p) : p \in X_\epsilon \} < \epsilon$$ for $x \in X$ I ...
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0answers
28 views

question in Numerical analysis

please guide me how to start and I will continue the another steps
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0answers
33 views

A question about $f(x)\equiv C$

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
2
votes
1answer
22 views

Help clearing up the definition of Limsup?

I was thinking about the equivalence of the two following definition of Limsup of a sequence. I find the definition 1 much more intuitive and I have been trying to convince myself of the equivalence ...
0
votes
0answers
7 views

Integral over homogeneous function does not vanish

Let $\alpha>0$ be a multi-index. For $x,y\in\mathbb{R}^n$, $n>1$, we consider the integral $$\int_{|x|=1} \int_{|y|=1} \partial_y^\alpha f(y)\ g(x,y)\ \mathrm{d}y \mathrm{d}x\qquad (*)$$ where ...
0
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1answer
33 views

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function. It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow ...
0
votes
1answer
18 views

Proof completeness of $L^p$

I'd like to check if I understood the proof that $L^p$ is complete ($1 \le p <+\infty$). I have to use the following fact: in a metric space, if a Cauchy sequence has a convergent subsequence then ...
0
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1answer
18 views

how to show boundary of either open or closed set is nowhere dense.

how to show boundary of either open or closed set is nowhere dense. i think we need to use baire category thm? countable intersection of dense, closed set is once again a dense, closed (and ...
1
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0answers
22 views

$\mu_x\otimes\mu_y$ and $\mu_y\otimes\mu_x$

Let $X$ and $Y$ spaces endowed with measures $\mu_x$ and $\mu_y$ defined on set semirings $\mathfrak{S}_x$ and $\mathfrak{S}_y$ and let $A\subset X\times Y$ be a subset of $X\times Y$ such that ...
1
vote
2answers
46 views

Does $\sum\limits_{k=1}^n a_k^2$ imply $\sum\limits_{l=1}^k a_k \in o(\sqrt{n})$?

I'm trying to determine some limits and it makes me wonder if my intuition about asymptotics is just wrong: Our calculus professor used to say that $\sum\limits_{n=1}^{\infty} \frac{1}{n}$ is ...
2
votes
3answers
44 views

Asymptotic behavior of $\sum\limits_{k=1}^n \frac{1}{k^{\alpha}}$ for $\alpha > \frac{1}{2}$

As the title states, I'm interested in the asymptotic behavior of $$\sum\limits_{k=1}^n \frac{1}{k^{\alpha}} , \alpha > \frac{1}{2}$$ for $n \to \infty $. Any hints/ideas?
0
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0answers
18 views

Composition of a compact support function with a increasing one

Let $u\in H^1(\mathbb{R}^n)$ have compact support and $c:\mathbb{R}\to\mathbb{R}$ is smooth, with $c(0)=0$ and $c'\geqslant 0.$ I am trying to prove $c(u(x))\in L^2(\mathbb{R}^n)$ or $c'(u(x))\in ...
0
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0answers
19 views

True or False: If $f$ is differentiable at $a$ and $g$ is differentiable at $f(a)$, then $(g\circ f)''(a)=g'(f(a))f''(a)+g''(f(a))(f'(a))^2$

True of False: If $f$ is differentiable at $a$ and $g$ is differentiable at $f(a)$, then $(g\circ f)''(a)=g'(f(a))f''(a)+g''(f(a))(f'(a))^2$. I wasn't sure if my interpretation of this problem was ...
4
votes
1answer
70 views

If $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function such that $f(x)=0$ for only finitely many values of $x$, [duplicate]

If $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function such that $f(x)=0$ for only finitely many values of $x$, then prove/disprove the following : $(a)$ Either $f(x) \geq 0 ~~\forall ...
1
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0answers
12 views

smallest σ-field on subset of Borel sets

If $A$ is a Borel subset of $R$, prove that the smallest σ-field of subsets of $A$ containing the sets open in $A$ is $\{C \in B(R):C \subset A\}$.
2
votes
1answer
41 views

Bolzano-Weierstrass application?

I am having problems proving the following claim: Given a bounded set $A \subset R^n$, I want to prove the existence of $a_1, \dots, a_N \in R^n$ and numbers $r_1, \dots, r_N \in [0, +\infty)$ such ...
0
votes
1answer
36 views

There exists a positive real number $u$ such that $u^3 = 3$

Modify the Theorem that states There exists a positive real number x such that $x^2 = 2$. Show that there exists a positive real number $u$ such that $u^3 = 3$. So far, I have come up ...
1
vote
1answer
64 views

Prove that if $ x_n = \sum_{k = 2}^n\sum_{m = 2}^k (m\ln{m})^{-1}$ then $ \lim(x_n)=\infty.$

Prove that $$\lim x_n=\infty$$ Hint: Show that $x_{2^k} \geq (2\ln2)^{-1}(1 + 2^{-1} +\ldots + k^{-1})$ I am absolutely confused at how to approach this question. If I could just receive a tip ...
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3answers
36 views

Uncountably many disjoint sets [on hold]

Show that if $\mu$ is a finite measure, there cannot be uncountably many disjoint sets $A$ such that $\mu(A)>0$
5
votes
6answers
972 views

How to show that the product of two irrational numbers may be irrational?

Show that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers. All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}^2 = 2$.
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votes
4answers
52 views

Example Real Analysis closed bounded [on hold]

Prove that if S is nonempty closed bounded subset of R then S has a minimum element. Don't know how to approach this or solve it. Help needed thank you!
2
votes
1answer
47 views

If functions converge a.e. and their integrals converge, does convergence in $L^1$ follow?

I was wondering if $f_n, f:\mathbb{R}\rightarrow\mathbb{R}$ are s.t. $f_n\rightarrow f$ pointwise a.e. and $\int f_n\rightarrow \int f$ where integrals are Lebesgue Integrals, is there any Theorem or ...
1
vote
2answers
29 views

Show :$\{B_n^2\}\to0\implies \{B_n\}\to0$.

How do I show that if $\lim B_n^2=0$ then $\lim B_n=0$? I tried to use a contradiction where I say suppose $B_n$ does not goes to zero. But then I realize that We cannot say $B_n$ goes to something ...
3
votes
3answers
45 views

Continuous surjective function from [0,1] to (0,1)

Is there a Continuous surjective function from [0,1] to (0,1)? I think there's none. Since we could take a sequence $f_n$ to approximate $1$ and by continuity and subjectivity then there's ...
2
votes
0answers
28 views

Square of uniformly convergent functions is not always uniformly convergent

For each $n ∈ N$ and $x ∈ R$ let $f_n(x) = x + \frac{1}{n}$, and for each $x ∈ R$ let $f(x) = x$. Working directly from the definition of uniform convergence (i.e. without using the Uniform Norm ...
0
votes
0answers
15 views

Showing that the norm of the image of a C1-function does not have a maximum.

Here's the setup: Let $U \subset \Bbb R^n$ be open and let $f: U \rightarrow \Bbb R^n$ be a $C^1$ function such that $f'(x)$ is invertible for all $x$ in $U$. Show that $x \mapsto ||f(x)||$ does not ...
1
vote
0answers
17 views

Limit of finite measure on a $\sigma$-field

let $\mu$ be a finite measure on the $\sigma$-field $F$. If $A_{n} \in F, n=1,2,...$ and $A=\lim_{n} A_{n}$, show that $\mu (A)=\lim_{n \to \infty} \mu(A_{n})$
1
vote
2answers
20 views

Question about the limsup of a sequence of real numbers

Let $\{p_n\}$ be a sequence of real numbers. Is it true that if $p_n$ diverges, i.e. $p_n \rightarrow \infty$, then $\limsup~p_n=\infty$? Here $\limsup~p_n$ is the supremum of the set of all ...
0
votes
1answer
44 views

Prove $A\rightarrow 0$ iff $A^2\rightarrow 0$

The implication from left to right is pretty easy. But the implication from right to left does not work. I tried to proof by contradiction by supposing $A^2$ converges to some number not 0 for the ...
3
votes
1answer
22 views

Showing that sequences of functions converge uniformly

Let $A$ be a set, let $f,g : A → R$, and let $(f_n)_{n∈N}$ and $(g_n)_{n∈N}$ be sequences of functions from $A$ to $R$. Show that if $f_n → f$ uniformly and $g_n→g$ uniformly, then $f_n + g_n →f + g$ ...
0
votes
1answer
16 views

Question about the upper limit of a sequence of real numbers

Let $\{p_n\}$ be a sequence of real numbers. I would like to show that $~lim_{n \rightarrow \infty}sup~p_n = lim_{n\rightarrow \infty}sup_{m \geq n} \{p_m\}$. But I am unclear on what is meant by the ...