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.

learn more… | top users | synonyms

0
votes
0answers
20 views

Proving a subset of $l_2$ is closed

Let $l_2$ be the set of all real sequences $x=(x_n)$ such that $\sum|{x_n}|^2 <\infty$ and define the norm $||x_n||_2=(\sum\limits_{n=1}^{\infty}|x_n|^2)^{\frac{1}{2}}$. I want to show that $A=\{ ...
1
vote
3answers
25 views

$\delta-\epsilon$ Question on Ordered Field $\mathbb{R}$

I got came across this question with the $\delta-\epsilon$ definition of a limit, but I do not know how to use it to solve the context of this problem: Problem: Let $f:\mathbb{R}\to\mathbb{R}$ be ...
2
votes
1answer
39 views

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing?

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing? In process of solving this problem, I faced to the problem of proving that $A::$: ...
0
votes
0answers
26 views

How do I prove this statement about the operator norm?

I stumbled across this equation in a paper, which may seem obvious, but I'm wondering if someone can explain why this is true? By definition of an operator norm, $$\left[(D^*D)^{-1} - ...
0
votes
1answer
18 views

Showing a function is Strictly Decreasing and find the right hand limit

$g(r)=e^{(1/r)(ln( \sum_{i = 1}^{n} |x_i|^r))}$, where all $x_k$ are non-zero. So I have to show that it's strictly decreasing on (0, $\infty$) $g'(r)= (e^{(1/r)(ln( \sum_{i = 1}^{n} ...
0
votes
2answers
28 views

Showing that stereographic projection is a homeomorphism

For any $n\geq 0$,the unit $n$-sphere is the space $S^{n}\subset \mathbb{R^{n+1}}$ defined by $$S^{n}=S^{n}(1) :=\left\{ (x_{1},...,x_{n+1}) \left\vert\,\sum_{i=1}^{n+1} x_{i}^{2}=1\right.\right\}$$ ...
3
votes
2answers
151 views

Is the product of two monotone sequences monotone?

Question: The product of monotone sequences is monotone, T or F? Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing. CASE I: Suppose ...
1
vote
1answer
17 views

Proving an identity using Riemann-Stieltjes Integration?

Prove the following identity using Riemann-Stieltjes Integration: $$\sum_{n=1}^N \frac{1}{n^s} =\frac{1}{N^{s-1}} + s \int_1^N \frac{\lfloor x\rfloor}{x^{s+1}}dx$$ Here's what I have so far: $$ ...
3
votes
1answer
55 views

Prove the sequence determined by $a_{n+1}={a_n\over \sin a_n}$ is convergent, and found its limit.

Let $\{a_n\}$ be a sequence defined by $0<a_1<{\pi \over 2}$, $a_{n+1}={a_n\over \sin a_n}$. $Attempt:$ $a_1>0$ and $\sin a_1>0$ and therefore the sequence begins positive and remains ...
4
votes
1answer
50 views

Evaluating $\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k} $

Question: How to compute $$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $$ Here is what I have tried so far: Define $s_n=\sum\limits_{k=1}^n \frac{1}{k 2^k}$ for every index $n$, ...
0
votes
1answer
10 views

What does Shilov mean by a “vector of absolute divergence”?

In Real and Complex Analysis, Problem 6-18, Shilov defines a vector of absolute divergence: ...
0
votes
0answers
4 views

How to verify if this is a autocovariance function?

Is this$$γ(h) = 1(h = 0) − 0.5 · 1(|h| = 2) − 0.25 · 1(|h| = 3)$$an autocovariance function? How to check this? Is there a method one can use to check if a given function is an autocovariance ...
2
votes
1answer
37 views

Applying the Stone-Weierstrass Theorem to approximate even functions

Let $f:[-1,1] \rightarrow \mathbb{R}$ be any even continuous function on $[-1,1]$ (i.e. $f(-x)=f(x)$ $\forall x \in [-1,1]$). Let $\epsilon>0$. Prove that there exists an even polynomial $p$ ...
0
votes
1answer
24 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
0
votes
2answers
31 views

Find the points where the function is continuous

Let $X \subset \mathbb{R}$ be a finite set and define $f: \mathbb{R} \to \mathbb{R}$ by $$ f(x)= \begin{cases} 1 & \text{if $x\in X$},\\ 0 &\text{otherwise}. \end{cases} $$ At which points ...
0
votes
0answers
20 views

Prove that any function f in $L^p$ is the limit, in the metric of $L^p$, of a sequence of simple functions. [on hold]

I know that I'll need to use dominated convergence here. In the problem, they ask to consider, when f is bounded and nonnegative, the sequence: $s_n(x) = \begin{cases} \frac{i-1}{2^n} \text{ for } ...
1
vote
1answer
23 views

Improper Integral - Multiple Choice Problem - $I$

Let $f$ be a function defined $\forall~ x\geq 1$.Let $n$ denote a positive integer and let $I_n$ denote the integral $\int_1^nf(x)dx$ which is always assumed to exist. Which of the following ...
0
votes
0answers
20 views

If $S_n=\sum_{k=1} ^n{}a_k \to \infty$ as $n\to \infty$ then $\sum_{k=1}^{\infty}{a_k\over {1+a_k}}$ diverges. [duplicate]

Let $a_n>0$ and let $S_n=\sum_{k=1} ^n{}a_k \to \infty$ as $n\to \infty$. Prove $\sum_{k=1}^{\infty}{a_k\over {1+a_k}}$ diverges. I am confused by this sort of sequence\sum thing. How can I use ...
0
votes
1answer
21 views

If $f_n \rightarrow f$ in $L^p$ and $g_n \rightarrow g$ in $L^q$, where $\frac{1}{p} + \frac{1}{q} = 1$, show that $f_n g_n \rightarrow fg$ in $L^1$ [duplicate]

I know that this will have something to do with Holder's inequality but I am at a loss as to how the $L^p$ and $L^q$ convergence in $f$ and $g$ dictate the convergence in $L^1$. Any help is ...
1
vote
4answers
83 views

Find $\lim_{n\to \infty}(\cos{x\over n})^{n^2}$

Find $$\lim_{n\to \infty}\left(\cos{x\over n}\right)^{n^2}$$ where $x\in \Bbb{R}.$ I tried using taylor series. A complete mess, and an area I am not very good at. I tried using $e$ which also gave me ...
3
votes
2answers
27 views

existence of a borel probability measure on $[0,1]$ such that $\int f d\mu=\lim_{k\to\infty}\frac {1} {N_k} \sum_{i=1}^{N_k}f(x_i)$ given sequence

Hi I'm really suck with this one, i would really appreciate it if any one can help me with this! prove that for $\{x_i\}\subseteq[0,1]$ there are $1\le N_1<N_2...$ and a probability measure $\mu$ ...
1
vote
4answers
70 views

Computing the sixth derivative of $F(x) = \int_1^x\sin^3(1-t)\mathrm dt$

Compute the sixth derivative at $x_0 = 1$ of $$F(x) = \int_1^x\sin^3(1-t)\mathrm dt$$ It's from a multiple choice test. I was able to narrow down the choices to $0$ and $60$. I guessed $0$ and ...
1
vote
1answer
27 views

Is this theorem about integration with substitution wrong?

A theorem in my book states: If $g$ is differentiable, f is continuous, and F is an antiderivative of f, then : $\int f[g(x)]g'(x)dx=F[g(x)]+C$ The reason I am asking if this is correct, ...
3
votes
2answers
23 views

Limit Definition for Proof

How can I use the definition of a limit to set up a proof for a statement such as: $$\lim_{n\to\infty} {f(n)} \to \infty$$ I have tried applying the standard definition, but I come out with ...
0
votes
2answers
21 views

Given two nonempty subsets A and B,

If A contained in B, and B is bounded above, show that A is also bounded above, and supA ≤ sup B. I know that supA ≤ b ∈ B but how do I show the first part?
5
votes
1answer
40 views

Show that $ \int_I x^{-\frac{1}{4}} $sin$ x \;dx \leq \pi^{\frac{3}{4}}$.

Let $I=[0,\pi]$. Show that $\displaystyle \int_I x^{-\frac{1}{4}} $sin$ x \;dx \leq \pi^{\frac{3}{4}}$. My Work: I think this is an application of Holders inequality. But any positive power of $ ...
0
votes
2answers
37 views

proving a limit by definition

given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a>1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition. proof: given $a_n=a^{\frac{1}{n}}$ for ...
3
votes
3answers
161 views

Inequalities proven by real analysis or induction.

Let $t\in [-1,1]$. Prove that $(1+t)^p+(1-t)^p\ge2$ when $p\ge 1$ and that $(1+t)^p+(1-t)^p \le 2$ where $0 \le p\le 1$. I am not sure how I should solve it. I tried induction at first and it was ...
1
vote
2answers
32 views

Find $\max, \min, \inf, \sup$

Find the $\max, \min, \sup, \inf$ if they exist of $B=\frac{n}{2n+1}$ such that $n$ is a natural number` I know that the $\min B=0=\inf B$, but I'm struggling to understand what the $\sup B$ and ...
0
votes
2answers
33 views

Show a sequence is bounded (therefore has convergent subsequence by Bolzano-Weierstrass)

I'm trying to show that this sequence is bounded (and hence, by B--W, has a convergent subsequence): $$ a_{n} = \frac{n\cos^n(n)}{(n^2+n)^{1/2}}$$ I noticed that the numerator is less than or ...
0
votes
4answers
65 views

Convergence of $\int_0 ^\infty \frac {dx}{\sqrt {1+x^3}}$

Convergence of $\int_0 ^\infty \dfrac {dx}{\sqrt {1+x^3}}$ Attempt: $\lim_{x \rightarrow \infty} \dfrac {x^{\frac{3}{2}}}{\sqrt {1+x^3}} =1$ Hence, $\dfrac {1}{x^{\frac{3}{2}}}$ and $\dfrac ...
1
vote
1answer
25 views

A question regarding a problem in Folland

Let $f_n(x) = ae^{-nax}-be^{-nbx}$ where $0<a<b$. Prove that $$ \sum_1^\infty f_n \in L^1([0,\infty),m)\quad\text{and}\quad\int_0^\infty\sum_1^\infty f_n(x)\,dx=\log(b/a). $$ This is the ...
0
votes
1answer
10 views

What is the connection between slant/oblique asymptote to the polynomial part of the function and polynomial division?

What is the connection between slant/oblique asymptote calculation to the polynomial part of the function and polynomial division? To find the slant asymptote $y=mx+n$ we can can calculate it in two ...
4
votes
2answers
42 views

$\lim_{n \rightarrow \infty} \frac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $

The value of $$\lim_{n=\infty} \dfrac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $$ Attempt: $S = \lim_{n \rightarrow \infty} \sum_{n=0} ^\infty \dfrac {k^{a+1}} {n.( 1^{a ...
2
votes
1answer
18 views

Adjoints of operators between different Hilbert spaces.

When we have an operator $$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$ from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint ...
1
vote
3answers
40 views

Union of two $\sigma$-algebras is not $\sigma$-algebra

Here is another very basic analysis problem but that puzzles me: Find an example of set $X$ and its two $\sigma$-algebras $\mathscr A_1$ and $\mathscr A_2$, such that $\mathscr A_1 \cup \mathscr ...
2
votes
2answers
31 views

Which of the following statements are true $(NBHM - 2015)$?

Let $X =\{f \in C[-5, 5] : f(-5)= f(5) = 0 \} $ There exist $ f \in X$ such that $ f \equiv 2$ on $[-1, 0]$ and $ f \equiv 3$ on $[1 , 2] \cup [3 , 4]$ For every $ f \in X$, there exist ...
1
vote
1answer
10 views

A set with non-$\sigma$-algebra monotone class

Working on this very basic analysis problem: Find an example of set $X$ with its monotone class $\mathscr M$ such that $\emptyset, X \in \mathscr M$, but it is not a $\sigma$-algebra. My ...
1
vote
0answers
14 views

Directed notion of openness

Is there a notion of openness which only considers certain directions? e.g. In $\mathbb{R^n}$, given some set of directions (i.e. vectors) $D = \left\{ d_1, \ldots, d_r \right\}$, a set $X$ is ...
0
votes
3answers
32 views

Show that Set $A = \{\frac{m}{m+n}:m,n \in \mathbb{N}\}$ is such that $\sup A=1$

Consider the set, $A = \{\frac{m}{m+n}:m,n \in \mathbb{N}\}$ I am confused at how to get $\sup A =1$ and was wondering if someone could explain the steps to me. Thank-you
1
vote
1answer
36 views

Can a divergent alternating series by rearrangement of terms be made to converge to a value?

Riemann discovered that a conditionally convergent series, through rearrangement of it's terms, can be made to converge to any value. But, if $S$ is a divergent alternating series, through ...
0
votes
1answer
32 views

Functions and continuity proof in real analysis

Prove: If $f\colon A\rightarrow\mathbb{R}^m$ and $a\in A$, show that $\lim_{x\rightarrow a}f(x)=b$ if and only if $\lim_{x\rightarrow a}f^i(x)=b^i$ for $i=1,\dots,m$. The end of the statement is ...
0
votes
4answers
56 views

Demonstrate that this equation has three real roots if fullfills this condition

I have this: demonstrate that this equation $$ {x}^{3}+p\,x+q=0 $$ has one real root if "p" is positive and has three real roots if $$ \frac{{q}^{2}}{4}+\frac{{p}^{3}}{27}<0 $$ I did ...
0
votes
0answers
20 views

Is it possible to show the uniqueness of formula for solution?

The motivation to this question can be found in: Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$ My question is: Is it possible to show the uniqueness of the formula for the ...
0
votes
1answer
45 views

How can the modulus of something be less than zero?

I've been asked to prove that for $\epsilon>0$, $$| a-x| < \epsilon \iff a-\epsilon<x<a+\epsilon,$$ and as a hint to consider both $| x-a|>0$ and $| a-x|<0$. I used the fact that ...
1
vote
1answer
36 views

Convergence of $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$

Convergence of $$\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$$ Attempt: $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1}) \sim \sum_{n=1}^\infty n^s( \sqrt n )$ As ...
0
votes
0answers
28 views

calculate $\lim_{n \to \infty}(\int_{a}^{b} f^nx(x)g(x)dx)^{\frac{1}{n}}$ [on hold]

Suppose that $g,f:[a,b] \to (0,\infty) $ are two continous functions, calculate $$\lim_{n \to \infty}\Bigg(\int_{a}^{b} f^n\ x (x)\ g(x) \ dx \Bigg)^{\frac{1}{n}}$$ where $f^n (x)= f(x)^n =f(x)\ ...
0
votes
1answer
28 views

Discontinuity of a function

Take $f(x)=\frac{1}{\sqrt{x}}$ on $(0;1)$ and 0 everywhere else on $\mathbb{R}$ take also an enumeration $\{r_n\}_{n\in\mathbb{N}}$ of the rationals on $(0;1)$. And define $g(x):=\sum_1^{\infty} ...
2
votes
2answers
34 views

Logic behind a proof in Topological Vector Spaces

I found the following result at the beginning of some notes on topological vector spaces (TVS). This is a quite fundamental result, that apparently is considered the corresponding version of the ...
1
vote
1answer
41 views

Interval of convergence of the series $\displaystyle\sum_{m=1}^{\infty}x^{\ln (m)}$ [on hold]

The series $\displaystyle\sum_{m=1}^{\infty}x^{\ln (m)}$ is convergent on the interval: (a) $(0,1/e)$ (b) $(1/e,e)$ (c) $(0,e)$ (d) $(1,e)$ I tried to apply the Cauchy condensation test, but I did't ...