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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with 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.

0
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0answers
7 views

Defining a functional over the set of all bounded functions

Let $B_{\mathbb{R}} = \{f: \mathbb{R} \to \mathbb{R}, \sup_{x\in\mathbb{R}}|f(x)| < \infty\}$ and $p:\mathbb{R} \to \mathbb{R}$ be $$p(f) = \inf_\sigma\limsup_{x\to\infty}\frac{1}{N}\sum_{k=1}^{N}f(...
0
votes
1answer
12 views

Sub-subsequences convergence

$(z_n)_n$ converges $\Rightarrow$ $\exists z \in \mathbb{C}$ such that every subseq. of $(z_n)_n$ has a convergent subseq. with limit $z$. Sorry, but I don't know how to start — would be amazing if u ...
1
vote
0answers
12 views

Weaker Definition for Hemi Continuity - Left continuous

A real function $f$ is continuous if for all convergence sequence $a_n$ and $b_n$ such that $b_n = f(a_n)$, $a_n\to a$, and $b_n\to b$, we have $b = f(a)$. A real correspondence $F$ is upper hemi ...
0
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0answers
24 views

Question about integration by parts when we have a sum of integrals.

I have the integral: $$\int_0^1f(x)dx=\int_0^1\int_0^1g(x,y)dxdy \:+\:\int_0^1h(x)dx $$ I must integrate by parts with respect to $x$ the left side, $\int_0^1f(x)dx$. Can I integrate by parts $\...
3
votes
1answer
21 views

Existence of continuous $r(t)$ with $\lim_{t \to \infty} \frac{f(r(t))}{g(t)} = 1$

Let $ \ f,g: \mathbb{R} \to \mathbb{R} \ $ be continuously differentiable functions such that $$\lim_{t \to \infty} f(t) = \infty = \lim_{t \to \infty} g(t) \ \ . $$ My question is: Is there a ...
1
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0answers
9 views

Uniform continuity of the function of two variables

Can you help me with this problem? Let $E\subseteq \mathbb{R}$, $U=\mathbb{R}^{2} \backslash E \times\{0\}$. Prove that the following statements are equivalent: 1)Any continious function $f:U\to\...
0
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0answers
17 views

prove $\ln(1+x^2)\arctan x=-2\sum_{n=1}^\infty \frac{(-1)^n H_{2n}}{2n+1}x^{2n+1}$

I was able to prove the above identity using 1) Cauchy Product of Power series and 2) integration but the point of posting it here is to use it as a reference in our solutions. other approaches ...
0
votes
2answers
28 views

Find sequences such that …

Let $c\in \mathbb{R}$. Find two sequences $(a_n)_n$, $(b_n)_n \subset \mathbb{R}$ with: (i) $\lim\limits_{n\to\infty}a_n=\infty, \lim\limits_{n\to\infty}b_n=0 $ and $\lim\limits_{n\to\infty}a_nb_n=c$ ...
0
votes
2answers
32 views

Show that if $X \subset R$ is bounded above, then $\overline{X}$ is also bounded above.

Show that if $X \subset R$ is bounded above, then $\overline{X}$ is also bounded above. Here $\overline{X} = \{a\in \mathbb{R} \text{ | if }a\text{ is an adherence point of }X \}$ My Proof by ...
1
vote
3answers
30 views

Determining if the set $\{e^{-x} : x\geq 0\}$ is compact or not.

Decide if the set $\{e^{-x} : x\geq 0\}$ is compact or not. If not compact give a counter example. Reasoning: So it is known that if a set is compact it is closed and bounded. The two possible ...
3
votes
5answers
85 views

Show that $a_{n+1}=2+\frac{1}{a_n}$ is convergent

Let $a_1 =1$ and $a_{n+1}=2+\frac{1}{a_n}$ where $n\in\mathbb{N}$. How can I show that the sequence $\left\{a_n\right\}$ is convergent? What is the limit? Please help me solve it. Thanks in advance....
4
votes
0answers
23 views

Decay of Fourier Transform of a Schwartz Function

Suppose we have a function $f(x)\in \mathcal{S}(\mathbb R)$; that is, it is a function in Schwartz space. Further, suppose we know that $$|f(x)|\leq Ce^{-|x|}.$$If it is helpful, we can actually ...
0
votes
2answers
41 views

If the integral of a contunuous function is positive does it implies that the function is positive

Let $f\in C[a,b]$. If we have for all $[c,d]\subset[a,b]$ $$\int_c^d f(x)dx\geq 0,$$ can we deduce that $f\geq 0 \; \forall \; t\in[a,b]$ ?
1
vote
1answer
29 views

Proof: Convergence of subsequences

$(z_n)_n$ converges $\Longleftrightarrow$ $\exists z \in \mathbb{C}$ such that every subseq. of $(z_n)_n$ has a convergent subseq. with limit $z$. I am completely stumped concerning this proof. Any ...
0
votes
1answer
25 views

Lower bounding $(1-2^{-n})^n$

I'm trying to prove the following statement: For each $n \in \mathbb{N}$, $(1-2^{-n})^n \geq \frac{1}{2}$ My attempts: I've first written $(1-2^{-n})^n=(\frac{2^n - 1}{2^n})^n=\frac{(2^n - 1)^n}{...
-1
votes
2answers
46 views

what do these equivalence sets look like? $x$~$y$ if $x-y$ rational,

For this equivalence relation, for $x,y \in R [0,1], x$~$y$ whenever $x-y$ is rational What do these classes look like? Is there one class that is all of the rational numbers, and then every ...
2
votes
3answers
45 views

Prove : For $k\in \mathbb{Z}: \lim\limits_{n\to\infty}\sqrt[n]{n^k}=1$

Proof: For $k\in \mathbb{Z}: \lim\limits_{n\to\infty}\sqrt[n]{n^k}=1$ Proof: Observe $k=1$. Then $\lim\limits_{n\to\infty}\sqrt[n]{n}=1$. Let $c_n:=\sqrt[n]{n}-1$, then: $$\begin{gather} n=(1+c_n)^n&...
1
vote
0answers
14 views

$(-r,r)\subset S_r$ for $r$ is sufficiently small if $E_r=E\cap (-r,r)$ and $S_r=\{2x-y:x,y\in E_r\}$

Let $E\subset (-\infty, \infty)$ be a Lebesgue measurable set which has the point $x=0$ as a point of density. ($|E_r|/(2r)\xrightarrow[r\to 0]{} 1$) For $r>0$, define $$E_r=E\cap (-r,r)\quad\...
0
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1answer
42 views

Showing that the set $\{e^{-x}: x > 0, x\in \mathbb{R}\}$ is neither open or closed rigourously.

I want to show that the set $\{e^{-x}: x > 0, x\in \mathbb{R}\}$ is neither open or closed rigourously. I have the idea intuitively which I will describe for each scenario below, but before that ...
0
votes
1answer
29 views

Showing that $ \lim_{n \to \infty} \sum_{x = 1}^{20} \cos(x-10)^{2n} = 1\ $

How do I show that $$ \lim_{n \to \infty} \sum_{x = 1}^{20} \cos(x-10)^{2n} = 1\ ?$$
-1
votes
0answers
38 views

Is L'Hôpital's method two-side?

Is it true that for differenced $f(x)$ and $g(x)$ such that $g(a)$ not zero, $\displaystyle\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}$ only if $\displaystyle\lim_{...
0
votes
1answer
23 views

proportion of the voters/ Central limit theorem

I want to compute the proportion of the voters p. Therefore I consider random variables $X_k$ for $k=1,...,n$: $$ X_k=\left\{\begin{array}{ll} 1, party \ is \ elected: "p" \\ 0, party \ ...
1
vote
2answers
30 views

If $\sum a_n$ converges absolutely, then so does $\sum f(a_n).$

Hi I am stuck on this problem. Let f: R to R be differentiable, with continuous derivatives and f(0) = 0 If $\sum{a_{n}}$ is an absolutely convergent series then $\sum{f(a_{n})}$ is also absolutely ...
1
vote
1answer
51 views

Proof: $\exists z\in \mathbb{C}$ such that every sub-sequence of $(z_n)_n$ has a convergent sub-sequence with the limit $z$.

Claim: Let $(z_n)_n$ be a sequence in $\mathbb C$ such that the subsequences $(z_{2n})_n$, $(z_{2n+1})_n$ and $(z_{3n})_n$ are convergent. Then there $\exists z\in \mathbb{C}$ such that every sub-...
14
votes
1answer
142 views

Prove that $\forall \epsilon > 0$: $\lim_{t\to\infty}t^{-2}\int_{0}^{t}[(f(x))^{1+\epsilon}/f'(x)]\,\mathrm dx =+\infty$

Let $f: [0,+\infty) \to [0,+\infty)$ be differentiable, $f' > 0$. Prove that $$\forall \epsilon > 0: \lim_{t\to\infty}\dfrac{1}{t^2}\int_{0}^{t}\dfrac{\left(f(x)\right)^{1+\epsilon}}{f'(x)}\...
0
votes
1answer
28 views

Prove images of injection are disjoint.

Let $A, B, C$ be sets such that $A \subseteq B \subseteq C$, and suppose that there is an injection $f : C \rightarrow A$. Define the sets $D_0, D_1, D_2, ...$ recursively by setting $D_0 := B \...
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0answers
23 views

f is continuous in R and limit x->infinity f(x) , limit x-> - infinity f(x) exists and is finite. Prove that f is UNIFORMLY CONTINUOUS [duplicate]

$f$ is continuous in $\Bbb{R}$ and $\lim_{x \to \infty} f(x)$ and $\lim_{x\to - \infty} f(x)$ exist and are finite. Prove that f is UNIFORMLY CONTINUOUS
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0answers
8 views

An application of Von Neumann's ergodic theorem

I am interested in the following problem Let $k\in \{1,2,\ldots,9\}$ and $N\in \mathbb N$. Let $a(N)$ be the number of integers $n=0,\ldots,N$ such that the number $2^n$ starts with the digit $k$ ...
-3
votes
0answers
16 views

Other than $\inf_{s \in [0,t]} f(s) = \inf_{s \in [0,t]} g(s)$ under what additional assumption $f(t) = g(t)$ for all $t$.

Consider two functions $f(\cdot)$ and $g(\cdot)$ such that for all $t$ $$\inf_{s \in [0,t]} f(s) = \inf_{s \in [0,t]} g(s)$$. Under what additional assumption $f(t) = g(t)$ for all $t$.
6
votes
0answers
27 views

From $\|g\|_2 =1$ to $\|g\|_\infty^2 \ge \dim V$ on a subspace of $C[0,1]$

Let $V$ be a finite dimensional subspace of $C[0,1]$. Prove that there exists $g \in V$ such that $\|g\|_2 =1$, $\|g\|_\infty^2 \ge \dim V$ In this problem we use the notation $$\|f\|_2 = \left(\...
6
votes
1answer
91 views

Show that the $n$-th Fibonacci number is given by $\frac{\cosh na}{\cosh a}$ or $\frac{\sinh na}{\cosh a}$, where $\sinh a=1/2$

This question is taken from book: Advanced Calculus: An Introduction to Classical Analysis, by Louis Brand. The book is concerned with introductory real analysis. I request to help find the solution. ...
3
votes
2answers
40 views

If $f$ is continuous, then $G$ is connected . True/false?

Let $X$ be a compact topological space and let $f : X \rightarrow \mathbb{R}$ be a function . The graph $f$ is the set $G = \{ (x, f(x) ) : x \in X \} \subseteq X \times \mathbb{R}$...
0
votes
0answers
9 views

Non-negative function greater than a polynomial whose derivatives are smaller than the function

Let $f:[0,\infty)\to \mathbb{R}$ no negative function. What functions do they comply with $|f^{(k)}(t)|≤C(k)f(t)$ and $t^n\leq Mf(t)$ for any $k>0$, and $t$ enough large, and $C(k),M>0$ ...
0
votes
1answer
23 views

Proving that a function is Riemann-Stieltjes integrable

Let $g$ a increasing function, and $f$ integrable with respect to $g$ in $J=[a,b]$ proof that $|f|$ is integrable with respect to $g$ By definition if $f$ is integrable with respect to $g$, for ...
6
votes
2answers
81 views

Find all roots of the equation :$(1+\frac{ix}n)^n = (1-\frac{ix}n)^n$

This question is taken from book: Advanced Calculus: An Introduction to Classical Analysis, by Louis Brand. The book is concerned with introductory real analysis. I request to help find the solution. ...
2
votes
1answer
35 views

$(Y,d)$ is a complete metric space $\rightarrow$ $(B(X,Y),p)$ is a complete metric space.

$(Y,d)$ is a complete metric space $\rightarrow$ $(B(X,Y),p)$ is a complete metric space. Where $B(X,Y)$ is the space of all bounded functions from $X$ to $Y$ and $p$ is the sup norm given by: $p(...
0
votes
0answers
14 views

Goodness of discrete approximation

I am given a function $f: R \times R \to R$ and a random variable $Y$ for which all finite moments exist. Suppose that $Y$ is a discrete random variable and it takes values $\mu_i$ with probability $...
-2
votes
0answers
25 views

How can I calcute an asymptote of integral? [on hold]

I have $f(x)=\int\limits_0^x g(t)dt$, and I need calculate an asymptote of $f(x)$ for $x\rightarrow\infty$. Do you know some ways to do this? I think that here can be useful Taylor series.
2
votes
4answers
79 views

The asymptotic behavior of $n\ln n -n$ [on hold]

How do I show that $$\displaystyle\lim_{n\to\infty}\dfrac{n\ln n - n}{\ln n!}=1?$$
0
votes
0answers
42 views

Prove for $k\in \mathbb{Z}$ that $\lim\limits_{n\to\infty}\sqrt[n]{n^k}=1$

Prove for $k\in \mathbb{Z}$ that $\lim\limits_{n\to\infty}\sqrt[n]{n^k}=1$ My attempt: Observe $k=1$, then $\lim\limits_{n\to\infty}\sqrt[n]{n}=1$. Let $x_n:=\sqrt[n]{n}-1$. Then: $$n=(1+x_n)^n>1+...
0
votes
3answers
35 views

Prove that $(z_n)_n$ is bounded $\Rightarrow$ $(w_nz_n)_{n\geq 1}$ is a null sequence (a sequence tending to $0$)

Let $z\in \mathbb{C}$, $(z_n)_{n\geq 1} \subset \mathbb{C}$ and $(w_n)_{n\geq 1}$(a null sequence) be sequences. Prove that $(z_n)_n$ is bounded $\Rightarrow$ $(w_nz_n)_{n\geq 1}$ is a null ...
0
votes
1answer
75 views

Proving the limit $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ [duplicate]

I want to prove that $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ There is a solution of the sum provided in my text book. There the expansion of $(1+\frac{1}{n})^n$ is like below: $(1+\frac{1}{n})...
2
votes
3answers
73 views

Calculate $\lim _{x \rightarrow 0} \frac{\int _{0}^{\sin x} \sqrt{\tan x} dx}{\int _{0}^{\tan x} \sqrt{\sin x} dx}$

Calculate $$ \lim _{x \to 0}\frac{\int_{0}^{\sin\left(x\right)} \,\sqrt{\,\tan\left(t\right)\,}\,\mathrm{d}t} {\int_{0}^{\tan\left(x\right)}\,\sqrt{\,\sin\left(t\right)\,}\, \mathrm{d}t} $$ I have ...
0
votes
3answers
60 views

Find $S_{n}=\sum_{k=1}^{n}k!(k^2+1)$

$$n\in\mathbb{N}^{*}; S_{n}=\sum_{k=1}^{n}k!(k^2+1)$$ I need to find $S_n$ I started like this: $S_{n}=\sum_{k=1}^{n}(k+2)!-3(k+1)!+2k!$ How to continue?I tried to give the k values but the terms ...
1
vote
0answers
18 views

Characterization of measurable functions mapping into banach space via simple functions?

This is a statement I read in Lang's Real and Functional Analysis Chpt VI, Sec 1, M8. M8. A map $f:X\to E$ with $E$ finite dimensional real vector space is measurable if and only if $f$ is a limit ...
0
votes
0answers
30 views

functions convergence

$f$ is bounded function defined around $x_0$. For every monotonic sequence $x_n \rightarrow x_0$, $f(x_n)$ is convergent. prove/disprove : all $f(x_n)$ sequences converge to the same limit when $x_n\...
0
votes
0answers
38 views

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is a function. And $f_{xx}$ and $f_{yy}$ exist and continuous. Is $f$ a twice differenced function?

Obviously, it is true if $\delta>0$ imply $2f(x+\delta, y+\delta)-f(x+2\delta, y)-f(x, y+2\delta)=o(\delta^2)$. But this result I can't prove too.
-1
votes
0answers
15 views

Find the equivalent of a sequence [duplicate]

let $u_n$ such as $u_0 > 0$ and $u_{n+1}=u_n+1/u_n$ Show that $u_n \sim \sqrt{2n}$
0
votes
0answers
53 views

Find limit of sequence defined by sum of previous terms and harmonics [on hold]

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function? I have been ...
0
votes
2answers
38 views

$S=\sum_{k=2}^{n}\frac{k^{2}-2}{k!}, n\geq 2$

$$S=\sum_{k=2}^{n}\frac{k^{2}-2}{k!}, n\geq 2$$ I got $S=\sum_{k=2}^{n}\frac{1}{(k-2)!}+\frac{1}{(k-1)!}-\frac{1}{k!}-\frac{1}{k!}$ I give k values but not all terms are vanishing.I remain with $\...