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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.

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1answer
22 views

Proof of Density of Rational Numbers as given in Howie

I'm working out of John M. Howie's Real Analysis (2001). In section 1.4 (Exercise 1.5), we're asked the following: Let $x$ and $y$ be real numbers, with $ x < y $. Show that, if $x$ and $y$ are ...
3
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3answers
75 views

Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
3
votes
1answer
45 views

Can one characterize the set of all $A\subseteq\mathbb{R}$ satisfying $2\cdot A\cdot A\subseteq A$ and $A\cdot(\mathbb{R}\backslash A)\subseteq A$?

This question is a spin-off of this question. When trying to solve that question, we came up with the idea of construction functions using sets $A\subseteq \mathbb{R}$ having the properties that $2xy\...
0
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1answer
30 views

Convergence of a sequence… [duplicate]

Let $\{a_n\}$ be a sequence of real numbers. Define $\sigma_n = 1/n(a_1 + \dots + a_n)$. Suppose that $\lim a_n = a \in \mathbb{R}$. Show that $\lim \sigma_n = a$. Here is my work so far... Fix $\...
0
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1answer
47 views

When is Taylor series a polynomial?

We know that Taylor series is a rational function if and only if its coefficients satisfy the linear recurrence relation. Can we put further conditions on the coefficients so that the Taylor series ...
2
votes
1answer
36 views

A polynomial whose roots form an arithmetic progression

Let $f$ be a fourth degree polynomial whose roots form an arithmetic progression. Prove that $f'$'s roots also form an arithmetic progression. I didn' t make much progress, I just wrote $f(x) =a(x-b-r)...
1
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2answers
51 views

Prove $f(x)= 1$ for $x \in \mathbb{Q}$ and $f(x)=0$ for $x \notin \mathbb{Q}$ is not integrable.

I want to prove that $$f(x) = \begin{cases} 1 \text{ for } x \in \mathbb{Q}\\ 0 \text{ for } x \notin \mathbb{Q} \end{cases}$$ is not integrable on $[0,1]$. Now I'm at the point in the book ...
-1
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0answers
45 views

Help calculate the limits

Help calculate the limits: 1) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{\sqrt {k(n-k)}}$$ 2) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{(n-k)\ln{n}} $$ 3) $$\lim_{n\to \infty}{n}^p \sin(\pi(\sqrt 2 ...
1
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0answers
35 views

Application of Hahn-Banach Theorem?

I am going to quote some versions of Hahn-Banach Theorem and try to deduce a statement which might be wrong. Thm 1.1 Let $E$ be a real normed vector space and $F\subset E$ as subspace. $\lambda:F\to ...
0
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0answers
23 views

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $\displaystyle\lim_{x\to\infty}f'(x)=0$. [duplicate]

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $\displaystyle\lim_{x\to\infty}f'(x)=0$. Prove that$\displaystyle\lim_{x\to\infty}\frac{f(x)}{x}=0$. I am trying to use cauchy ...
3
votes
1answer
109 views

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. Show $\lim_{n \rightarrow \infty} x_{n}$ exists. [duplicate]

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. show $\lim_{n \rightarrow \infty} x_{n}$ exists. To do this the problem has been broken down into three pieces: a) Show that $x_{n} <...
1
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1answer
32 views

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence?

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence? My attempt via induction: If I prove that the denominator grows faster than the numerator, I can conclude ...
1
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0answers
30 views

Convergence in measure on a finite and infinite measure space

Let $(E,\mathcal{A},\mu)$ be a measure space. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions, which converges in measure to $f:\forall \epsilon>0,\lim_n\mu(\left\{|f_n-f|>\...
0
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2answers
73 views

what is $\lim_{x\to 0} f(x)$

Let $$f(x)=\sum_{n=1}^{\infty}{\sin(nx)\over n^2}$$ Then what is $\lim_{x\to 0} f(x)$ Now I know the series converges uniformly by $M-Test$ (Take $M_n=1/n^2$). What should be my next step. I am ...
-2
votes
3answers
92 views

How do I solve $\int_0^1(x-1)^2(1-x)dx$? [on hold]

How do I solve $\int_0^1(x-1)^2(1-x)dx$?
0
votes
1answer
14 views

Condition upon integral divergence

I have had no classes in analysis and was wondering if the following is true, and if so, how one proves it. Proposition: Let $f$ be a continuous function along the domain $(a,b)$. Let $f(a)$ be ...
-4
votes
2answers
54 views

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ [on hold]

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ We have $-1\leq\cos{x}\leq 1$. So $(\cos{x})^n \to 0$ as $n\to \infty$ Please solve this problem. Please find the sum.
0
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1answer
15 views

Calculating derivative with multiple variables

Let z = f(x,y), x = x(t,s) and y = y(t,s) all be twice continously differentiable functions Try to find $$\frac{\partial z^2}{\partial t^2}$$ I've tried it and only got: $$\frac{\partial z}{\...
2
votes
1answer
24 views

For a triangular array, if $\max_{1\le k\le n}x_{n,k}\to 0$ and $\sum _{k=1} ^nx_{n,k}\to \lambda$ does $\prod_{k=1}^n(1+x_{n,k })\to e^{\lambda}$?

Consider the following: we have a triangular array of nonnegative numbers $$x_{1,1 } \\x_{2,1 } \ x_{2,2 } \\ x_{3,1 } \ x_{3,1 } \ x_{3,3 } \\... $$ The maximum on each row converges to zero: $\...
0
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0answers
42 views

Examples of increasing homeomorphism from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities

Let $\varphi : \mathbb{R}_+ \to \mathbb{R}_+$ be an increasing homeomorphism satisfying $\varphi(0)=0,$ where $ \mathbb{R}_+:=[0,\infty).$ For example, $\varphi(s)=\frac{s^3}{1+s^2}$ for $s \in \...
1
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0answers
17 views

prove $ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$

$$ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$$ for all $u \in H^1(\Omega)$ for the open unit circle $\Omega$ in $\mathbb{R^2}$. ...
0
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0answers
23 views

Best bound for the remainder of two variables Taylor's theorem

Let $f:\mathbb{R}^2 \to\mathbb{R}$ be a two times differentiable function. Then Taylor theorem says that there exists $t\in [0,1]$ such that \begin{align*} f({\bf x})=f({\bf a})+\sum_{i=1}^{2}\frac{\...
0
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0answers
36 views

Thinking process behind proving $\exists$ solution to $f(x) = x$ when f is bounded

Real Analysis; prove $f(x)=x$ has at least one solution. Here is a solution. I proved initially by doing contradiction, assuming $g(x) = x$ , $\forall x\in$ domain of f, either $f(x) > g(x)$ or $f(...
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0answers
46 views

‎If ‎$‎\lim‎ f(x) = L‎$ as ‎$‎x‎\rightarrow ‎+‎\infty‎$‎, then $‎\lim‎ f(c x) = L‎$ as $‎x‎\rightarrow ‎+‎\infty‎$.

‎Let ‎‎$‎f‎$ ‎be a‎ ‎real-variable ‎function ‎such ‎that ‎‎$‎‎‎‎‎\lim‎‎ f(x) = L‎$ as ‎$‎x‎‎\rightarrow ‎+‎\infty‎$ where $‎L\in‎\mathbb{R}‎$‎. ‎Also, let‎ $‎c‎>0‎$ be a constant. My question is:‎ ‎...
0
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1answer
25 views

About Baire's theorem

Exercise $3.22$ of baby Rudin is to prove the Baire's theorem: Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove Baire's theorem, ...
0
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2answers
37 views

Prove that $f: X \rightarrow R$ is continuous with respect to the metric $d_1$ on $X$ iff it is continuous with respect to the metric $d_2$ on $X$.

Let $X$ be a set and $d_1, d_2$ be metrics on $X$ so that for constants $m,M > 0$ and any $x,y \in X$ we have $md_1(x,y) \leq d_2(x,y) \leq Md_1(x,y)$ Prove that $f: X \rightarrow \mathbb{R}$ is ...
1
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2answers
46 views

A problem on number of solutions of a functional equations

Find all functions $f:R \rightarrow R$ such that $f(0)=1$ and for all $x\neq -1$ : $f(x)=8f(2x+1)$ (I have found only one solution: $1/(x+1)^3$. Method was by iterated substitution of $2x+1$ ...
2
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1answer
34 views

Compactness when mapping into a higher $L^p$ space and then back

Question: Let $q>p \ge 1$ and let $T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $i\circ T$ ...
0
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0answers
42 views

Uniformly Bounded and Bounded Variation [on hold]

Studying functions of bounded variation, the following exercise came up: Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of functions with $f_n:I \to \mathbb{R}$. Show that: If $(f_n)_{n\in \mathbb{N}}$...
6
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1answer
265 views

Proof verification: At most countably many local maxima

I'd appreciate a second pair of eyes on a proof. I want to prove that a function $f:\mathbb{R}\to\mathbb{R}$ can have at most countably many strict local maxima. The question has been asked elsewhere ...
0
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1answer
55 views

Differentiability of $|x|^p$?

Let $p > 0$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined piecewise by $f(x)= |x|^p$ if $x \in \mathbb{Q}$ and $f(x)=0$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. For what values of $p$...
1
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0answers
43 views

Let $f:[a,b] \rightarrow \mathbb{R}$ is cts with $f(a), f(b) < 0, \int_a^b f(x)dx \ge 0$. Show $\exists c \in (a,b) s.t. \int_c^b f(t) dt \ge 0$.

My solution to this problem is the following: Solution: $[a,b]$ is closed interval, so $f$ is uniformly continuous on the interval $[a,b]$. So, we know that there exists $\delta >0$ such that ...
1
vote
1answer
19 views

Set construction confusion

Let $S$ be an infinite bounded subset of $\mathbb R$. Now let's construct a set $T$ where, $$T=\{x:x \textrm{ exceeds at most a finite number of elements of } S\}.$$ Is there any element of $T$ which ...
2
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2answers
43 views

Let $(X,d)$ be a metric space and $ A\subseteq X$ be compact. Prove that for any $y \in X$ there exists $x \in A$ so that $d(y,x) = d(y,A)$ [duplicate]

Let $ (X,d) $ be a metric space and $ A \subseteq X $ be compact. Prove that for any $ y \in X $ there exists $ x \in A $ so that $d(y,x) = d(y,A)$ where $ d(x,y)=|x-y| $. Since A is compact it is ...
0
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2answers
45 views

Find the closure and the interior of A

I have to find the closure and the interior of the set A defined by $A =${$(x,\sin(x^{-1})) : x \in R-\{0\}$} $\subset$ $R^2$ I don't know how to start. I know that $\sin(x^{-1})$ has it's maximum ...
0
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0answers
8 views

Uniform convergence and integrals when domain is not a compact set.

Suppose sequence of continuous functions ${f_n}$ that converges uniformly to a continuous function $f$ on a closed interval $[a,b]$, then we have $$\lim_{n\to\infty}\int_a^b f_n(x) dx = \int_a^bf(x) ...
0
votes
0answers
32 views

If $\sum a_n$ is convergent but not absolutely, then $\sum a_n^+$ diverges

Let $a_n \in \mathbb{R}$, such that $\sum_{n=1}^\infty|a_n|= \infty$ and $\sum_{n=1}^m a_n \to a$, as $m \to \infty$. Let $a_n^+=\max\{a_n,0\}.$ Show that $\sum_{n=1}^\infty a_n^+= \infty$. Approach: ...
6
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4answers
195 views

If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$

Let $g(x)\ge0$. If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0,$ where $f$ is any integrable function. If simeone is allowed to use the Mean Value thorem for integrals, the proof is at hand. ...
0
votes
1answer
30 views

Proving Borel measurability of a function.

Suppose $f: \mathbb{R}^2 \to \mathbb{R}$ is a function such that $f(x, \cdot)$ is Borel-measurable for each $x$, and $f(\cdot, y)$ is continuous for every $y$. Define $f_n: \mathbb{R}^2 \to \mathbb{...
1
vote
1answer
41 views

Where did this definition come from?

Let $C,D > 0$. We call a function $f : \Bbb R \to \Bbb R$ pretty if $f$ is a $\Bbb C^2$-class, $|x^3 f(x)| \leq C$ and $|xf''(x)| \leq D$. (i). Show that if $f$ is pretty, then, given $\...
0
votes
1answer
62 views

Why is $\lim_{t\to\infty}\frac{(f(t))^{1+\varepsilon}}{tf'(t)}=\infty$ for all $f:[0,\infty)\to[0,\infty)$ such that $f'>0$? [on hold]

Let $f:[0,\infty)\to[0,\infty)$ be a differentiable function such that $f'(t)>0$ for all $t$ and let $\varepsilon>0$. Why is $$\lim_{t\to\infty}\frac{(f(t))^{1+\varepsilon}}{tf'(t)}=\infty\quad?...
0
votes
1answer
68 views

Is this product strictly positive? [on hold]

Let $p\in (0,1)$ and $\varepsilon\in (0, 1)$ be fixed. For all $i\in \mathbb N$ we define $p_i=1+(p-1)(1-\varepsilon)^i$. Is it possible to prove that $$\prod_{i=1}^{+\infty}\frac{p_i}{2-p_i}>0$$? ...
0
votes
3answers
65 views

I am stumped on this problem

It is from Intro to Analysis by Bartle 3rd ed Let $I:=[a,b]$ let $f:I\rightarrow\mathbb{R}$ be a continuous function with $f(x)>0$ Prove that there exists a number $c>0$ such that $f(x)\geq c ...
1
vote
2answers
34 views

Linearity property of integral involving a bounded linear operator

Suppose $X$ is a Hilbert space and $T\in\mathcal{B}(X)$. For $f\in\mathcal{C}([a,b],X)$, where $a\leq b$, we have \begin{equation} Tf\in\mathcal{C}([a,b],X)\quad\text{and}\quad T\int_{a}^{b}f(x)\,\...
0
votes
0answers
10 views

Set of roots Hermite polynomials(probabilistic type)

What is the nature of the set of all roots of the Hermite polynomials? They’re known to be all real. Is it a dense set? If not, what are the limit points?Are the limit points also roots? Are the limit ...
1
vote
1answer
42 views

What is $\int_0 ^1 \int_x^1 \frac{f(t)}{t} dt dx $ if $\int f =1$?

Let $f$ be a Lebesgue integrable function on $[0,1]$ and $\int f = 1,$ and let $$g(x) = \int_x ^1 \frac{f(t)}{t} dt \quad x \in [0,1].$$ Calculate the integral of $g$. I feel like I'm supposed to ...
1
vote
0answers
16 views

Sequence of bounded variation functions that converge uniformly to a function with unbounded variation

I need help with this exercise: Prove that $\exists (f_{n})_{n=1}^{\infty}\subset BV([0,1])$ such that $f_{n}\to f$ uniformly, but $\|f_{n}-f\|_{BV}\not\to 0$. So I proposed the sequence $(f_{n})_{n=...
5
votes
1answer
320 views

What does it mean for something to be strictly less than $\epsilon$ for an arbitrary $\epsilon$?

Perhaps a trivial question, but something I never completely understood. If we have shown that $a-b < \epsilon$ for all $\epsilon > 0$, then does that imply that $a-b \le 0$? I"m interested in ...
0
votes
1answer
14 views

Radius of convergence comparison for power series

Given two power series $g (x)=\sum a_n x^n $ and $h (x)=\sum b_n x^n$ with radius of convergence $R_1$ and $R_2$ such that $g(x) \leq h (x)$ for all $x \in \{|x| \leq min \{R_1,R_2\}\}$, Does this ...
0
votes
0answers
15 views

The behaviour of functions nested under themselves outwith their domains

This question follows from some interesting observations on a sum of reciprocals. Instead of summing them however, we will place each fraction to make a continued fraction. Some visualisations on ...