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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1answer
13 views

Interval of convergence of the series $\sum\limits_{m=1}^{\infty}x^{\ln (m)}$

The series $\displaystyle\sum_{m=1}^{\infty}x^{ln (m)}$ convergent on the interval (a) $(0,1/e)$ (b) $(1/e,e)$ (c) $(0,e)$ (d) $(1,e)$ I tried to apply Cauchy condensation test, but i cant get ...
0
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0answers
5 views

Properties of signed measure

In studying signed measures, I have a question. Let $\mu$ be a signed measure on sigma algebra $\mathfrak{M}$. How can I show that $\mu^+(E)=\sup\{\mu(F):F\subset E, F\in\mathfrak{M}\}$ and ...
1
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0answers
35 views

Proe that If $a_n>0$ and $\sum a_n$ converges then $\sum (\frac {b_n}{a_n})$ converges

Let $\{a_n\},\{b_n\}$ be two sequences such that for each $n$ we have $e^{ a_n }= a_n + e^{b_n }$. Show that if $a_n>0$ and $\sum a_n$ converges $\implies \sum (\dfrac {b_n}{a_n})$ converges ...
2
votes
2answers
20 views

Limit evaluation with integral

Evaluate the limit $$\lim_{n\to\infty} \int_0^1 n^2x(1-x^2)^n dx$$ My Proof: We may look at $n$ as a constant and evaluate the integral $\int_0^1 x(1-x^2)^ndx$ (I already moved out the $n^2$). ...
0
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1answer
10 views

Direct sum and intersection of sets

If $A_1,A_2,B_1,B_2$ are subspaces of a Hilbert space $H$, is the following statement true or not? $$\left( A_1 \oplus A_2\right)\cap \left( B_1 \oplus B_2\right)=\left(A_1\cap B_1\right)\oplus ...
0
votes
2answers
19 views

Checking when an $a$-dependent function is continuous, differentiable.

For some $a\in \Bbb{R}$ define a function $f_{a}(x) = \begin{cases} {x^{a}\cos{1\over x}}, & \text{if $x$ $\ne$ 0} \\[2ex] 0, & \text{if $x=0$} \end{cases}$. Hints firstly are preferred. b. ...
1
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2answers
32 views

Antiderivative of $\frac{\sqrt{4-x}}{x\sqrt{x}}$

I need help to find the antiderivative of the function $\displaystyle x \, \mapsto \, \frac{\sqrt{4-x}}{x\sqrt{x}}$ on $]0,4[$. I have tried the change of variables $u = \sqrt{4-x}$ but it didn't ...
0
votes
1answer
27 views

Determine when $f_{a}(x)$ is bounded.

For some $a\in \Bbb{R}$ define a function $f_{a}(x) = \begin{cases} {x^{a}\cos{1\over x}}, & \text{if $x$ $\ne$ 0} \\[2ex] 0, & \text{if $x=0$} \end{cases}$ What should $a$ be in order for ...
0
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0answers
19 views

e Online source for alternative proofs

I'm looking for some alternative proofs for various theorems. My goal is to compile a list of various proofs each relating to a specific theorem (such as the triangle inequality, Fermat's Little ...
0
votes
3answers
19 views

$X_n$ has limit $L$. Given a constant $C$ we have a new sequence $Y_n$ where for each $n,Y_n=X_n+C$. What is the limit of $Y_n$? [on hold]

$X_n$ has limit $L$. Given a constant $C$ we have a new sequence $Y_n$ where for each $n$, $Y_n=Xn+C.$ What is limit of $Y_n$? I need help with this problem; I don't understand how to approach it.
1
vote
1answer
12 views

positivity of a function

Consider the function $G(\omega):=\omega^4+2r^2\omega^2+b$, where $r,b\in\mathbb{R}$. What constraints on the parameters are needed to guarantee the positivity of $G$ for all $\omega>0$?
1
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2answers
22 views

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$.

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$. What I have: Assume that $k<0$, need to show that this contradicts the ...
0
votes
2answers
32 views

Show that for each $p \in P(n)$, the set $A(p) = \{x \mid p(x) = 0\}$ is countable

Show that for each $p \in P_n\;$, the set $\;A(p) = \{x \mid p(x) = 0\}\;$ is countable.   Where $P_n$ is the set of all polynomials of degree $n$ with integer coefficients. I just proved that ...
0
votes
3answers
93 views

Do I need to memorize everything? & How advanced do I need to learn in other subjects? [on hold]

After gaining a lot knowledge in Mathematics, esp. Real Analysis, my inclination is to start doing research, I mean not a lame research, I mean some research with new, revolutionary, profound ...
1
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2answers
35 views

A uniformly convergent sequence of real analytic functions which does not converge to a real analytic function

I am looking for an example of a uniformly convergent sequence of real analytic functions which does not converge to a real analytic function. Also I would appreciate any pointers on how to think of ...
2
votes
2answers
48 views

Is complement of a dense set in $\mathbb{R}$ dense in $\mathbb{R}$?

$\mathbb{Q}$ is dense in $\mathbb{R}$. Also, its complement, $\mathbb{R-Q}$, is dense in $\mathbb{R}$. I know that we can proof denseness of $\mathbb{Q}$ and $\mathbb{R-Q}$ separately for each of ...
3
votes
0answers
52 views

Infinite sum of analytic function still analytic

Consider $$ f_n(x) = n e^{-n^6(x-n)^2} : \mathbb R \rightarrow \mathbb R$$ and the series $$ f(x) = \sum_{n=1}^{\infty} f_n(x). $$ Is $f$ analytic on $\mathbb R$? A function is analytic if for ...
1
vote
2answers
31 views

Proving a sequence converges using epsilon-N definition.

I'm stuck with what to do next in my homework problem please help.
1
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0answers
23 views

Is a dense and co-dense subset $G_\delta$ or co-$G_\delta$

Let $A \subset \mathbb{R}$ such that $A$ and $A^C$ are both dense. By Baire's Theorem at most one of $A$ and $A^C$ is $G_\delta$ (i.e. a countable intersection of open sets) I couldn't think of an ...
-3
votes
1answer
24 views

Show that the sequence $\{b_j\}$ given by $b_j = j$ as $j$ approaches infinity is not bounded [on hold]

Show that the sequence $\{b_j\}$ given by $b_j = j$ as $j$ approaches infinity is not bounded by using the definition of boundedness of a sequence. Help please.
-2
votes
0answers
23 views

Real analysis open and closed set [on hold]

Please help me with the below question: Prove that the set $(0, 1)$ is open, $[0, 1]$ is closed, and $(0, 1]$ is neither open nor closed.
2
votes
0answers
18 views

Prove or disprove $\nu(E)=\lambda(f(E))$ is a measure provided that $f$ is nondecreasing and satisfies the N-condition.

Suppose $f$ is a non-decreasing continuous function from $[a,b]$ to $\mathbb{R}$, and $\lambda$ is the Lebesgue measure in $\mathbb{R^1}$. Also, $f$ satisfies the property that $f$ maps Lebesgue ...
2
votes
1answer
24 views

What variables is $\delta$ dependent on in the epsilon-delta definition of continuity?

The definition of continuity is: $f$ is continuous at $a$ if: Given any $\epsilon>0 $, $\exists \delta > 0$ st. $|x-a|<\delta \implies |f(x)-f(a)|< \epsilon$ $\delta$ obviously depends ...
1
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0answers
20 views

Rationals in an interval $[a,b] \in \Bbb R$

(i) For which real values $a$ and $b$, ($a < b$), is the set $[a,b] \cap \Bbb Q$ open in $(\Bbb Q, d)$, (where $d(x,y)= \lvert x-y \rvert$)? (ii)For which real values $a,b$ is the set $[a,b] \cap ...
2
votes
1answer
37 views

Proving standard properties of sine and cosine defined by their power series

Definition: We define $\displaystyle \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{\left ( 2n+1 \right )!}, \; x \in \mathbb{R} $ and $ \displaystyle \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n ...
0
votes
1answer
11 views

determine outer normal unit vector of $\{(x,y,z)|y^2+z^2\leq1\}$

I want to calculate the outer normal unit vector $n$ for the boundary of $$ A=\{(x,y,z)|x^2+y^2+z^2\le 1,x\ge0\} $$ So I have $\partial A=\{(0,y,z)|y^2+z^2\le1\}\cup\{(x,y,z):x^2+y^2+z^2=1,x\ge0\}$. ...
1
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0answers
13 views

Understanding the behaviour of $F(x)=e^{(1/(x-b)-(1/(x-a))}$

I'm trying to understand the behaviour of the function $F(x)=e^{(1/(x-b)-(1/(x-a))}$ over the open subset $(a,b)$ of $R^{1} $and Zero otherwise ( we let $0<a<b$ ). Is this function ...
2
votes
1answer
29 views

$f\in L^p(X,\mu)$ , $f-1\in L^q(X,\mu)$ then $\mu(X) < \infty $

Can some one give a hint how to start to solve : Assume $ 1 \le p,q < \infty $ and $$f\in L^p(X,\mu)$$ now if we assume $$f-1\in L^q(X,\mu)$$ then we have $$\mu (X) < \infty $$ Thanks If ...
2
votes
0answers
34 views

Is this a Hilbert space? If not, is it reflexive?

Let $E$ be a Banach space. Let $L^2(\Omega, E)$ denote the space of random variables taking values in $E$ with second order moment. Is $L^2(\Omega,E)$ a Hilbert space? or at least, reflexive? 1) I do ...
2
votes
3answers
35 views

If $\sum a_n$ converges absolutely , then so does, $\sum \frac {a_n^2} {1+a_n^2}$

If $\sum a_n$ converges absolutely , then so does, $\sum \dfrac {a_n^2} {1+a_n^2}$ Attempt: Given that $\sum a_n$ converges absolutely $\implies \sum |a_n|$ converges. ...
2
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0answers
16 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
-1
votes
1answer
16 views

How do I find the convergence of this summation using the comparison test? (∑(1/√(n^3-n)))

How do I find the convergence of this summation using the comparison test? \begin{equation} \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3 - n}} \end{equation} I am not sure what the comparison sequence ...
0
votes
1answer
21 views

calculate riemann integral on the triangle [on hold]

Calculate Riemann Integral $$ \int \int_{B} e^{4y^2}dxdy $$ where B is a triangle: (0,0), (0,1), (-1,1) I have no idea what should I do with that integral.
0
votes
1answer
40 views

Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges. [duplicate]

Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges. Attempt: $\sum |a_n|$ converges $\implies \sum |a_n|<M$. If $\sum |a_n|$ converges, then $\sum a_n $ also converges. ...
1
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2answers
59 views

Baby Rudin theorem 2.41: need a rigorous construction when $E$ is not closed.

In Baby Rudin (Principles of Mathematical Analysis), theorem 2.41 says, 2.41. If a set $E$ in $R^k$ has one of these three properties, it has the other two: $(a)$ $E$ is closed and ...
3
votes
2answers
51 views

convergence of $ \sum_{n=1}^{\infty} (-1)^n \frac{2^n \sin ^{2n}x }{n } $

Find values of $x$ for which the following series converges $$ \sum_{n=1}^{\infty} (-1)^n \dfrac{2^n \sin ^{2n}x }{n } $$ Attempt: (a) Check for Absolute Convergence If we consider $ ...
1
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1answer
35 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
1
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0answers
19 views

Invertibility condition for $f:\mathbb{R}^2\to\mathbb{R}^2$ on the domain boundary

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected domain $D\subset\mathbb{R}^2$ with a smooth boundary $\partial D$. I am interested in the local invertibility of $f$ in a ...
1
vote
1answer
33 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
1
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1answer
25 views

Derivative of bilinear forms

I want to solve the following problems: Let $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form. Prove that it's differential is $$ Df_{(x,y)}(a,b) = f(x,b) + f(a,y).$$ ...
-3
votes
0answers
34 views

Prove the inequality between the arithmetic and geometric mean

Assume that for $x_1,...,x_n\geq0$ we let $G=(x_1x_2\dots x_n)^{1/n}$ and $A=(x_1+x_2+...+x_n)/n$. I would like to know if the following procedure leads to a proof of $$G\leq A$$ The equality is ...
-1
votes
1answer
27 views

Prove that $(0,1)\times (0,1)$ is open in $\mathbb R^2$. [on hold]

Consider a plane $\mathbb R^2$ with the metric $$d(x, y) = \sqrt{|x_1 - x_2|^2 + |y_1 - y_2|^2}.$$ Show that $U = (0,1) \times (0,1)$ is an open set in $\mathbb R^2$ under this metric. How to ...
1
vote
1answer
33 views

Determine if a function is a metric

I have been asked the following question in one of my tests. I'm not sure of how to do it. Consider the plane $X = \Bbb R^2$. For each of the following two proposed distance functions, determine ...
0
votes
0answers
11 views

Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv ...
1
vote
0answers
18 views

Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
2
votes
2answers
36 views

$f : ]0, \infty[\to\mathbb{R}$ Prove that $\lim_{x\to\infty} f(x) = 0$ if $\lim_{x\to\infty} [f(x)+f'(x)] = 0$

I tried $ f(x)=\frac {f(x)e^x}{e^x} $ but now I'm stuck. How is it possible to apply the $\frac{\infty}{\infty}$ - LHR?
1
vote
1answer
41 views

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: ...
-3
votes
1answer
15 views

bounded and connected bounded interval are equivalent [on hold]

X is a subset of real line.Then the following statements are equivalent (a) X is bounded and connected. (b) X is a bounded interval.
0
votes
2answers
18 views

The infimum $\inf_{(a,b) \in A\times B} \; \rho(a,b)$ is attained for any two compact sets $A,B$

Let $A,B$ be compact sets in $(S,\rho)$. Define $\rho(A,B)$ by $$\rho(A,B) = \inf_{(a,b) \in A\times B} \; \rho(a,b)$$ Show that there exists $a_0 \in A, b_0 \in B$ s.t. $$\rho(A,B) = \rho(a_0,b_0)$$ ...
3
votes
1answer
47 views

Infinity in “Extended Natural Numbers”

In Baby Rudin, p. 27, it is stated that the $\infty$ in notations like $\sum\limits_{i=0}^\infty i$ and $\bigcup\limits_{i=0}^\infty A_i$ is not the same as the $+\infty$ in the extended real ...