# Tagged Questions

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, integration through the fundamental theorem of calculus...

6answers
4k views

### How can you prove that a function has no closed form integral?

I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction ...
11answers
3k views

### Zero to the zero power - Is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...
5answers
3k views

### Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
3answers
938 views

### Set of continuity points of a real function

I have a question about subsets $$A \subseteq \mathbb R$$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize ...
2answers
712 views

### bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$

could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you.
3answers
2k views

### Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
4answers
1k views

### Sine function dense in $[-1,1]$

We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
8answers
2k views

### Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$

To prove the convergence of $$\sum_{n=1}^{\infty} \frac1{n^p}$$ for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test. I am wondering if there is a ...
2answers
1k views

### Convergence a.e. and of norms implies that in Lebesgue space

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why is it ...
3answers
2k views

### Construction of a Borel set with positive but not full measure in each interval

I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere. To be precise, if $\mu$ denotes Lebesgue ...
7answers
993 views

### Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?

Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$ I.e - does the second half of the harmonic series go to zero? I know that for a finite number of terms the limit of the sum is ...
3answers
718 views

### Continuity of the metric function

Let $(X,d)$ be a metric space. How to prove that for any closed $A$ a function $d(x,A)$ is continuous - I know that it is even Lipschitz continuous, but I have a problem with the proof:  |d(x,a) - ...
2answers
4k views

### How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$

If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?
4answers
2k views

### How do I define a bijection between $(0,1)$ and $(0,1]$?

How do I define a bijection between $(0,1)$ and $(0,1]$? Or any other open and closed intervals? If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if ...
4answers
1k views

### Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
3answers
566 views

### A vector space over $R$ is not a countable union of proper subspaces

I was looking for alternate proofs of the theorem that "a vector space $V$ of dimension greater than $1$ over an infinite field is not a union of fewer than $|\mathbf{F}|$ proper subspaces" and ...
5answers
1k views

### Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$

As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem ...
11answers
2k views

### $\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite

How do I prove that $\displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
1answer
1k views

### Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$

Here is an exercise, on analysis which i am stuck. How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on ...
3answers
2k 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 ...
5answers
945 views

### Perfect set without rationals

Give an example of a perfect set in $\mathbb R^n$ that does not contain any of the rationals. (Or prove that it does not exist).
4answers
1k views

### Limit of the nested radical $\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$ [duplicate]

Possible Duplicate: On the sequence $x_{n+1} = \sqrt{c+x_n}$ Where does this sequence converge? $\sqrt{7},\sqrt{7+\sqrt{7}},\sqrt{7+\sqrt{7+\sqrt{7}}}$,...
5answers
865 views

### $\lim_{n\rightarrow\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\rightarrow\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. ...
2answers
283 views

### Methods to evaluate $\int _{a }^{b }\!{\frac {\ln \left( tx + u \right) }{m{x}^{2}+nx +p}}{dx}$

Today I saw a question with an answer that made me rethink of the following question, since it's not the first time I try to find an answer to it. If you look at the answer of Mhenni Benghorbal here ...
3answers
958 views

### Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
2answers
923 views

### Does uncountable summation, with a finite sum, ever occur in mathematics?

Obviously, “most” of the terms must cancel out with opposite algebraic sign. You can contrive examples such as the sum of the members of R being 0, but does an uncountable sum, with a finite sum, ...
3answers
556 views

### Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums. ...
7answers
1k views

### Is there any geometric way to characterize $e$?

Let me explain it better: after this question, I've been looking for a way to put famous constants in the real line in a geometrical way -- just for fun. Putting $\sqrt2$ is really easy: constructing ...
5answers
3k views

### Does $\int_0^{\infty}\frac{\sin x}{x}dx$ have an improper Riemann integral or a Lebesgue integral?

In this wikipedia article for improper integral, $$\int_0^{\infty}\frac{\sin x}{x}dx$$ is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ...
4answers
3k views

### Continuous bijection from $(0,1)$ to $[0,1]$

Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map.
5answers
975 views

### Proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$

Thomson et al. provide a proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$ in this book. It has to do with using an inequality that relies on the binomial theorem. I tried to do an alternate proof ...
1answer
1k views

### Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
3answers
433 views

### When $L_p = L_q$?

As we know that $L_p \subseteq L_q$ when $0 < p < q$ for probability measure, I was wondering when $L_p = L_q$ is true and why. Is it to impose some restriction on the domain space? Thanks!
2answers
785 views

### examples of measurable functions on $\mathbb{R}$

Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
3answers
877 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 ...
2answers
643 views

### What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$?

This question was motivated by another question in this site. As explained in that problem (and its answers), if $\displaystyle f$ is continuous on $\displaystyle [0,1]$ and \$\displaystyle \int_0^1 ...
3answers
233 views

### uniform convergence of few sequence of functions

Pick out the sequences $\{f_n\}$ which are uniformly convergent. (a) $f_n(x) = nxe^{−nx}$ on $(0,∞)$. (b)$f_n(x) = x^n$ on $[0, 1]$. (c)$f_n(x) = \frac{\sin(nx)}{\sqrt{n}}$ on $\mathbb{R}$. (d) ...
1answer
302 views

### If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?

Let $f(x)$ be a continuous real-valued function on $[a,b]$ and $M=\max\{|f(x)| \; :\; x \in [a,b]\}$. Is it true that: $$M= \lim_{n\to\infty}\left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n} ?$$ ...
1answer
379 views

### continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$

Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$. $g(0)=1$. If $a=g(1)$,then $a>0$ ...
1answer
1k views

### How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
4answers
4k 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 ...
3answers
1k views

### Can there be two distinct, continuous functions that are equal at all rationals?

Akhil showed that the Cardinality of set of real continuous functions is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree ...
4answers
1k views

### Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
2answers
844 views

### Comparing the Lebesgue measure of an open set and its closure

Let $E$ be an open set in $[0,1]^n$ and $m$ be the Lebesgue measure. Is it possible that $m(E)\neq m(\bar{E})$, where $\bar{E}$ stands for the closure of $E$?
3answers
327 views

### A log improper integral

Evaluate : $$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$ I found it can be simplified to $$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$ I found the exact value in the ...
3answers
534 views

### $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx$ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would ...
5answers
175 views

### Given $y_n=(1+\frac{1}{n})^{n+1},n \in \mathbb{N},n \geq 1$ Show that $\lbrace y_n \rbrace$ is a decrasing sequence

Given $$y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.$$ Show that $\lbrace y_n \rbrace$ is a decrasing sequence. Anyone can help ? I consider the ratio ...
4answers
428 views

### Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$

I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrasingly weak - I would like to show that, for a real number \$p ...
1answer
732 views

### how to show this is a dense set?

I am not able to prove that this set is dense in $\mathbb{R}$. Will be pleased if you help in a easiest way, $\{a+b\alpha: a,b\in \mathbb{Z}\}$ where $\alpha\in\mathbb{Q}^c$ is a fixed irrational.
2answers
465 views

### limit of $\frac{a_{n+1}}{a_n}$

Today my lecturer put up on the board that: If $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}$ exists and $a_n>0$ then \$\displaystyle ...
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