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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2
votes
2answers
997 views

A converse of sorts to the intermediate value theorem, with an additional property

I need to solve the following problem: Suppose $f$ has the intermediate value property, i.e. if $f(a)<c<f(b)$, then there exists a value $d$ between $a$ and $b$ for which $f(d)=c$, and also ...
3
votes
3answers
931 views

Exercise 9.2 from Apostol's Mathematical Analysis book. Uniform convergence of product

This is a problem (Exercise 9.2) from Apostol's Mathematical Analysis (second edition) which I couldn't solve. $\bullet$ Define two sequences $\{f_{n}\}$ and $\{g_{n}\}$ as follows: $f_{n}(x) = x \...
26
votes
3answers
3k views

Norms on C[0, 1] inducing the same topology as the sup norm

This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying. Let $C[0, 1]$ denote the ...
3
votes
2answers
865 views

Corollary to Baire's Category Theorem

In Rudin's Real and Complex Analysis (p. 97 in my 3rd edition), the following is stated as a corollary to Baire's Category Theorem: "In a complete metric space, the intersection of any countable ...
2
votes
2answers
96 views

How to show that $f(x)= \lim_{k \to \infty} \Bigl(\lim_{j \to \infty} \cos^{2j}(k!\cdot \pi \cdot x)\Bigl)$

How to show that: $$f(x)= \lim_{k \to \infty} \Bigl(\lim_{j \to \infty} \cos^{2j}(k!\cdot \pi \cdot x)\Bigl)$$ is the Dirichlet's function.
7
votes
1answer
988 views

function is smooth iff the composition with any smooth curve is again smooth

I'm stuck on the following part of a proof: Let $\phi: \mathbb R^m \to \mathbb R^n$ be a function such that $\gamma'(t) := \phi(\gamma(t))$ is smooth for every smooth function $\gamma: \mathbb R ...
5
votes
1answer
257 views

Decimal representation series

Given the following sequence $d(n) = x \cdot n - \lfloor x \cdot n \rfloor$ with $n \in \mathbb{N}$, $x \in \mathbb{R} \backslash \mathbb{Q}$ . I want to show that for every $z \in [0,1)$ there is a ...
3
votes
1answer
938 views

When is an infinite sum of polynomials analytic?

Consider the following function: $ f(x) = \sum_{i,j>=0} a_{ij} x^i (1-x)^j$ , where the coefficients satisfy $a_{ij} \in [0,1]$. Observe that $f(x)$ converges for $x\in [0,1]$. Is $f$ analytic over ...
5
votes
2answers
562 views

Solving a tough limit

I am trying to verify for which $z \in \mathbb{R}$ the series $\sum _{n=1}^{\infty } \left(1-\cos \left(\frac{1}{n}\right)\right)^z$ converges. The only test that was successful for me is the Kummer ...
6
votes
3answers
314 views

Can a norm only be defined on vector spaces?

All examples I have come across so far deal with norm defined on a vector space. Can norm only be defined on vector spaces?
28
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3answers
2k views

Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers

A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with non-...
11
votes
3answers
5k views

Does the sum of reciprocals of primes converge?

Is this series known to converge, and if so, what does it converge to (if known)? Where $p_n$ is prime number $n$, and $p_1 = 2$, $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$
43
votes
2answers
1k views

Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$

Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 ...
10
votes
2answers
238 views

What is the minimum value of $a$ such that $x^a \geq \ln(x)$ for all $x > 0$?

This is probably just elementary, but I don't know how to do it. I would like to find the minimum value of $a$ such that $x^a \geq \ln(x)$ for all $x > 0$. Numerically, I have found that this ...
36
votes
9answers
35k views

Proving that the set of limit points of a set is closed

From Rudin's Principles of Mathematical Analysis (Chapter 2, Exercise 6) Let $E'$ be the set of all limit points of a set $E$. Prove that $E'$ is closed. I think I got it but my argument is a bit ...
5
votes
3answers
391 views

Stochastic integral and Stieltjes integral

My question is on the convergence of the Riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. Here ...
5
votes
1answer
434 views

How many Borel-measurable functions from $\mathbb{R}$ to $\mathbb{R}$ are there?

How many Borel-measurable functions from $\mathbb{R}$ to $\mathbb{R}$ are there? The motivation is from this answer of mine on MathOverflow.
8
votes
2answers
5k views

Continuity of a function at an isolated point

Suppose $c$ is an isolated point in the domain $D$ of a function $f$. In the delta neighbourhood of $c$, does the function $f$ have the value $f(c)$?
7
votes
1answer
323 views

Arithmetic on $[0,\infty]$: is $0 \cdot \infty = 0$ the only reasonable choice?

On page 18 of Rudin's Real and Complex analysis he defines $0 \cdot \infty = 0$ and says that "with this definition the commutative, associative, and distributive laws hold in $[0,\infty]$ without any ...
5
votes
1answer
174 views

invariance of dimension under diffeomorphism of real subspaces

This is a problem from Arnold's book on ODEs I cannot solve. Prove that if $f:U\to V$ is a diffeomorphism, then the Euclidean spaces with the domains $U$ and $V$ as subsets have the same dimension. ...
0
votes
4answers
985 views

Prove the existence of a largest integer less than or equal to a rational number

Prove that there is a largest integer $n$ such that $n \le x$ for any fixed rational $x$. What about any fixed real number $x$?
4
votes
2answers
8k views

Basic question about proving uniqueness

I just started real analysis. I don't have a background in proofs or logic, simply calculus. So I'm trying to learn more about proofs--so forgive the basic question, please. How do you go about ...
20
votes
1answer
620 views

Spreading points in the unit interval to maximize the product of pairwise distances

This is prompted by question 15312, but moved to the reals. It must be solved already. Pick n points $x_i \in [0,1]$ to maximize $\prod_{i < j} (x_i - x_j)$. A little playing shows you don't ...
8
votes
1answer
491 views

Does the integral test work on higher dimensions?

The integral test of convergence states that, if $f:[1,+\infty)\to[0,+\infty)$ is a monotonically decreasing nonnegative function, then the series $\sum_1^\infty f(n)$ converges iff $\int_1^\infty f(n)...
7
votes
2answers
9k views

Is the natural log of n rational?

It's famously unknown whether the natural log of 2 is rational or not. How about the natural log of other numbers. Is it known/unknown whether these are rational? Obviously ln(1) is 0, and ln(2^n) ...
5
votes
2answers
4k views

Reverse Taylor Series

Almost everyone is familiar with the famous Taylor Series: $ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n $ which, if it converges at more than one point, will converge in some interval ...
9
votes
1answer
3k views

Convergence tests for improper multiple integrals

For improper single integrals with positive integrands of the first, second or mixed type there are the comparison and the limit tests to determine their convergence or divergence. There is also the ...
141
votes
6answers
45k views

When can you switch the order of limits?

Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \...
1
vote
0answers
224 views

Proving a statement related to norms in euclidean spaces

Taken from Rudin's Principles of Mathematical Analysis, 3rd edition (Chapter 1, Exercise 16): Suppose $k\geq 3, x,y\in\mathbb{R}^k, |x-y|=d>0$, and $r>0$. Prove: (a) If $2r > d$, there are ...
44
votes
4answers
10k views

Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?

I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of $P$...
8
votes
2answers
2k views

Dedekind Cuts versus Cauchy Sequences

Are there any advantages or disadvantages in defining a real number in the following ways: Definition 1 A real number is an object of the form $\lim\limits_{n \to \infty} a_n$ where $(a_n)_{n}^{\...
0
votes
2answers
140 views

Existence of factors of a complex number

Suppose $z$ is a complex number. Prove that there exists an $r \geq 0$ and a complex number $w$ (with $|w| = 1$) such that $z = rw$. Does $z$ uniquely determine $r$ and $w$? Let $z = a+bi$. Then $|...
1
vote
2answers
256 views

Complex Numbers and Square Roots

Suppose $z = a+bi$ and $w = u+iv$. Let $\displaystyle a = \left(\frac{|w|+u}{2} \right)^{1/2}$ and $\displaystyle b = \left(\frac{|w|-u}{2} \right)^{1/2}$. Show that $z^2 = w$ if $v \geq 0$ and $(\bar{...
4
votes
2answers
991 views

Logarithm and Uniqueness

Fix $b>1, y>0$ and prove that there is a unique real $x$ such that $b^x = y$. So for uniqueness, $x_1 < x_2 \Rightarrow b^{x_1} < b^{x_2}$. Then consider the set $S = \{b^t: t \leq x \}$ ...
2
votes
1answer
584 views

Exponents and Suprema

Fix $b>1$. Show the following: (a) If $m,n,p,q$ are integers, $n>0, q>0$, and $r = m/n = p/q$, prove that $(b^{m})^{1/n} = (b^p)^{1/q}$. So $(b^r) = (b^m)^{1/n}$. (b) Prove that $b^{r+s} = ...
3
votes
2answers
2k views

Supremum and Infimum

Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that $\inf A = -\sup(-A)$. We know that $-A$ is bounded above. Hence ...
23
votes
4answers
4k views

No maximum(minimum) of rationals whose square is lesser(greater) than $2$.

Suppose $A$ is the set of all rational numbers $p$ such that $p^2 <2$ and $B$ is the set of all rational numbers $p$ such that $p^2 > 2$. We want to show that $A$ contains no largest element and ...
0
votes
1answer
526 views

non increasing function

I am trying to understand a proof, I don't understand one of the steps. could someone shed some light why the following is true. if $f(x)$ is a non-increasing and continuous function and $a>b$ ...
3
votes
1answer
266 views

Term-by-term integration under a certain condition

The recent post here has led to the following question (consider $\cos(sx) = \sum\nolimits_{k = 1}^\infty {\frac{{( - 1)^k (sx)^{2k} }}{{(2k)!}}}$). I find it instructive, whether it will turn out to ...
13
votes
1answer
500 views

If $g^{-1} \circ f \circ g$ is $C^\infty$ whenever $f$ is $C^\infty$, must $g$ be $C^\infty$?

Suppose that $g$ is a bijection on the real line, and $g^{-1} \circ f \circ g$ is a $C^\infty$ function whenever $f$ is $C^\infty$. It seems howlingly obvious that this can only happen if $g$ is ...
10
votes
1answer
1k views

Is a uniformly continuous function vanishing at $0$ bounded by $a|x|+c$?

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be uniformly continuous with $g(0)=0,c\geq 0, c \in \mathbb{R}$. Show: $$\exists a\geq 0 \in \mathbb{R}: \forall x \in \mathbb{R}: |g(x)| \leq a \cdot |x|+c$$...
3
votes
3answers
363 views

Can $\ln(x)=\lim_{n\to\infty} n\left(x^{\frac1{n}}-1 \right)$ be expressed as an infinite telescoping product?

Concerning this question of mine which involves telescoping in the solution, I was wondering if it is possible to express $$\ln(x)=\lim_{n\to\infty}n\left(x^{\frac1{n}}-1\right)$$ and $$e^x=\lim_{n\...
0
votes
1answer
172 views

Reversing a convergent sequence

I've always had the following silly(?) doubt about convergent sequences. Given a finite sequence $A_{1:n}=(a_1,a_2\ldots ,a_n)$, define its reverse as $rev(A) = (a_n,a_{n-1},..a_1)$. Further, if $...
1
vote
1answer
447 views

Generalized Binomial Expansion of $\left(1+x \right )^{y}$

I was trying to expand $\displaystyle \left(1+x\right)^{y}$ as a power series in terms of $y$ using the Generalized Binomial Theorem, $\displaystyle \left(1+x\right)^{y}=\sum_{k=0}^{\infty}\binom{y}{...
6
votes
2answers
365 views

Does the fact that every interval in $\mathbb{R}$ is connected implies that $\mathbb{R}$ is order-complete?

Suppose that every open interval in $\mathbb{R}$ is a connected set. Does this implies the least upper bound axiom? (i.e every non-empty subset of $\mathbb{R}$ which is bounded above has a least upper ...
2
votes
1answer
375 views

recursively defined trigonometric sequence and Cesàro means

So, we have a function $$f(x) = \frac{3\sin(x)}{2 + \cos(x)} $$ and we fix a point $$ x_0 \in \left(0, \frac{2\pi}{3}\right] $$ and define a sequence by setting $$ x_{n+1} = f(x_n) $$ Now I need ...
2
votes
2answers
457 views

Can a linear operator from a normed space to itself be extended with operator norm preserved? [closed]

Let $X$ be a subspace of $\ell_1^4$ (i.e. $\mathbb{R}^4$ equipped with the $\ell^1$ norm). Can one always extend a linear operator $l:X\rightarrow \ell_1^4$ to $L:\ell_1^4\rightarrow \ell_1^4$ such ...
5
votes
2answers
1k views

Don't Understand Proof in Rudin Principles of Mathematical Analysis Book Thm 7.24?

I was rather confused about this proof I came across in Rudin Chapter 7. The premise is: If K is a compact metric space, if $f_n \in C(K)$ for $n = 1, 2, 3, ...,$ ($C(K)$ being a set of complex-...
16
votes
4answers
18k views

How to prove that the derivative of Heaviside's unit step function is the Dirac delta?

Here is a problem from Griffith's book Introduction to E&M. Let $\theta(x)$ be the step function $$\theta = \begin{cases} 0, & x \le 0, \\ 1, & x \gt 0. \end{cases} $$ The ...
6
votes
2answers
3k views

Is $\sum \sin{\frac{\pi}{n}}$ convergent?

I have to test for convergence of the series: $\displaystyle \sum\limits_{n=1}^{\infty} \sin\Bigl(\frac{\pi}{n}\Bigr)$ What i did was \begin{align*} \sin\Bigl(\frac{\pi}{n}\Bigr)+ \sin\Bigl(\frac{...