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.

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17
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2answers
502 views

Compute: $\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}} $

Compute the following limit: $$\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}} $$ I'm interested in almost any approaching way for this limit. ...
17
votes
1answer
1k views

Lebesgue measure of the graph of a function

Let $f:R^n \rightarrow R^m$ be any function. Will the graph of f always have Lebesgue measure zero? 1) I could prove that this is true if $f$ is continuous. 2) I suspect it is true if $f$ is ...
17
votes
3answers
378 views

Local maxima of Legendre polynomials

When I plotted the (normalized) Legendre polynoials, I couldn't help noticing that all the local maxima lay on a really nice curve: What is the equation of the curve (and how can we arrive to that ...
17
votes
1answer
918 views

A smooth function's domain of being non-analytic

I am wondering how much a smooth function may be non-analytic, because in proofs, whilst there non-analytic smooth functions, it would suffice if a smooth function were analytic on only a "small set". ...
17
votes
1answer
2k views

Example of a Borel set that is neither $F_\sigma$ nor $G_\delta$

I'm looking for subset $A$ of $\mathbb R$ such that $A$ is a Borel set but $A$ is neither $F_\sigma$ nor $G_\delta$.
17
votes
4answers
414 views

Existence of the limit

Let $f:\mathbb{R}\to\mathbb{R}$. The following limit exists for all $x \in \mathbb{R}$: $$\lim_{n\to ∞} f(nx) ,$$ where $n \in \mathbb N$. Is it correct that: $$\lim_{x\to ∞} f(x) ,$$ exists if: ...
17
votes
6answers
4k views

Video lectures on Real Analysis?

One of the most annoying gaps in my math education is Real analysis. I tried hard, but all I could find are either Harvey Mudd College lectures or MathDoctorBob. The latter are too short and the ...
17
votes
3answers
549 views

Find $f$ where $f'(x) = f(1+x)$

Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function such that $$f'(x) = f(1+x)$$ How can we find the general form of $f$? I thought of some differential equations, but not sure how ...
17
votes
1answer
782 views

$\sigma$-algebra in Riesz representation theorem

Let $X$ be a locally compact Hausdorff space and $I$ - a positive linear functional on $C_c(X)$. Then according to the Riesz representation theorem there exist a $\sigma$-algebra $\mathfrak{M}$ in $X$ ...
17
votes
4answers
430 views

What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$ Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to ...
17
votes
1answer
353 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 ...
17
votes
1answer
308 views

Boundedness of functions

Recently in my problem solving class we had the question: If $f: [0,\infty) \to \Bbb R$ is a twice differentiable function such that $f''(x) + e^x f(x) = 0$, then prove that $f$ is bounded. ...
17
votes
5answers
513 views

Proving a certain map on the closed unit disc must be the identity

Bounty expired. Will gladly re-create one if a satisfactory answer is posted in the future. Prove: Let $f$ be a continuous function on the closed unit disc with two properties: 1. $f$ is the ...
17
votes
4answers
806 views

Build $\mathbb{R}$ from $\mathcal{P}(\mathbb{N})$

It's well known that $\mathbb{R}$ has the same cardinality as $\mathcal{P}(\mathbb{N})$; but I would fain know if there is a way to construct $(\mathbb{R}, +,\cdot, \leq )$ using only definitions that ...
17
votes
1answer
213 views

Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$

Suppose $f:\mathbb{R}\to\mathbb{R}$ has period 1, and for some $q\in(0,1)$: $$|f(x)-f(y)|\leq q|x-y|\quad \forall x,y$$ Let $g(x)=x+f(x)$, for any $a\in\mathbb{R}$,define the following sequence: ...
17
votes
1answer
189 views

How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?

I want to prove the inequality $$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$ There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
16
votes
8answers
2k views

How I can prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2?

Prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2 My attempt I proved that the sequence is increasing and bounded by 2, can anyone help me show that the ...
16
votes
6answers
2k views

Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
16
votes
3answers
1k 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 ...
16
votes
4answers
427 views

Computing $\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$

What ways would you propose for the limit below? $$\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$$ Thanks in advance for your suggestions, hints! Sis.
16
votes
8answers
711 views

Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?

Here is my question. Why do the reals need to be "constructed" by this bizarre "Dedekind cut" or "equivalence class of Cauchy sequences" argument? Why can't they simply be "observed" as consisting ...
16
votes
2answers
1k views

Why can't Cantor sets cover $\mathbb{ R}$?

The Cantor set is uncountable so I expect countably many of them to be able to cover $\mathbb R$, but the set has measure $0$ so countably many of them also has set of measure $0$ and thus can't cover ...
16
votes
7answers
5k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...
16
votes
1answer
2k views

Enumerations of the rationals with summable gaps $(q_i-q_{i-1})^2$

Here is a question from my undergraduate days which I never knew the answer to. I just want to know if anyone can offer me a hint. Consider the rationals in $[0,1]$. Does there exist a (bijective) ...
16
votes
4answers
510 views

Is every open subset of $ \mathbb{R} $ uncountable?

Is every open subset of $\mathbb{R}$ uncountable? I was crafting a proof for the theorem that states every open subset of $\mathbb{R}$ can be written as the union of a countable number of disjoint ...
16
votes
11answers
860 views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
16
votes
3answers
471 views

PV $\int_0^\infty \frac{\log \cos^2 \alpha x}{\beta^2-x^2} \, dx=\alpha \pi$

$$ I:=PV\int_0^\infty \frac{\log\left(\cos^2\left(\alpha x\right)\right)}{\beta^2-x^2} \, dx=\alpha \pi,\qquad \alpha>0,\ \beta\in \mathbb{R}.$$ I am trying to solve this integral, I edited and ...
16
votes
5answers
4k views

Roots of Legendre Polynomial

I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials. Are the roots always simple (i.e., multiplicity $1$)? ...
16
votes
3answers
541 views

An interesting sum to infinity

Is there any simple way of computing the following sum? $$\sum_{k=1}^\infty \frac1{k\space k!}$$
16
votes
2answers
2k 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 $E$?
16
votes
3answers
705 views

Evaluate $\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$

Evaluate $$\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$$
16
votes
3answers
486 views

Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$

I wonder what kind of functions satisfy $$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$ I suppose all functions must be continuous.
16
votes
2answers
389 views

Proving that $(b_n) \to b$ implies $\left(\frac{1}{b_n}\right) \to \frac{1}{b}$

In my textbook (S. Abbott. Understanding Analysis 1 ed. pp 47 Theorem 2.3.3.iv), the author proves $b_n \to b$ implies $\frac{1}{b_n} \to \frac{1}{b}$ the following way: ...
16
votes
4answers
1k views

Why does Cantor's diagonal argument yield uncomputable numbers?

As everyone knows, the set of real numbers is uncountable. The most ubiquitous proof of this fact uses Cantor's diagonal argument. However, I was surprised to learn about a gap in my perception of the ...
16
votes
5answers
770 views

Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx = \frac{\pi^3}{16}-3G\log 2 \tag{1}$$ where $G$ is Catalan's Constant. I was able to express it in terms of ...
16
votes
1answer
334 views

Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
16
votes
3answers
345 views

The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$. My attempt : f is $2\pi$ periodic and $\mathcal{C}^3$, we have : ...
16
votes
6answers
1k views

If $f$ is uniformly differentiable then $f '$ is uniformly continuous?

The following theorem is true? Theorem. Let $U\subset \mathbb{R}^m$ (open set) and $f:U\longrightarrow \mathbb{R}^n$ a differentiable function. If $f$ is uniformly differentiable $ ...
16
votes
1answer
289 views

Prove the following integral inequality

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: ...
16
votes
2answers
2k views

How to show that a set of discontinuous points of an increasing function is at most countable

I would like to prove the following: Let $g$ be a monotone increasing function on $[0,1]$. Then the set of points where $g$ is not continuous is at most countable. My attempt: Let ...
16
votes
3answers
261 views

Intuition for the Universal Chord Theorem

So the Universal Chord theorem is a statement and proof that; The numbers of the form $r = \displaystyle \frac{1}{n} \ \ n \ge 1$ are the only numbers such that for any continuous function ...
16
votes
1answer
298 views

How to show $E:=\{x\in [0,1]: |f^{-1}(\{x\})|\,\mbox{ is even} \}$ is countable?

Hi everybody I have seen the following question which I could not solve it, so I thought I can share the question with you and ask for help. Question: Let $f:[0,1]\to [0,1]$ be a continuous function ...
16
votes
2answers
518 views

Wiggly polynomials

I'd like to be able to construct polynomials $p$ whose graphs look like this: We can assume that the interval of interest is $[-1, 1]$. The requirements on $p$ are: (1) Equi-oscillation (or ...
16
votes
1answer
461 views

A curious theorem by Peano

Let $f$ be defined on $[a,b]$ and there differentiable. Show that for every $ \epsilon>0 $ there exists a partition $\, a=a_0<a_1<...<a_n=b \,$ of $ \,[a,b] \,$ so that $$\left|\frac ...
16
votes
2answers
605 views

How does one prove that a multivariate function is univariate?

The question resembles How does one prove that a multivariate function is constant? but appears to be more difficult. Suppose that $u\colon\mathbb R^2\to\mathbb R$ is a continuous function such that ...
16
votes
1answer
305 views

Calculus over $\mathbb{Q}$

The mismatch between the sensitivity of 'mathematical calculus' and the flexibility of 'real world calculus' has been bothering me a bit recently. What I mean is this: in the real world, I can trust ...
16
votes
1answer
456 views

“Converse” of Taylor's theorem

Let $f:(a,b)\to\mathbb R$. We know that for every $c\in(a,b)$ we can write $f(t)=\sum_{i=0}^k a_i(c)(t-c)^i+o\left((t-c)^k\right)$ and $\forall i$ $a_i(c)$ is continuous (with respect to $c$). Can we ...
16
votes
1answer
429 views

Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?

I've written the question first, then the motivation behind it and lastly some background. Note that the question makes references to definitions and theorems written in the background bit at the end. ...
15
votes
5answers
887 views

Prove $|e^{i\theta} -1| \leq |\theta|$

Could you help me to prove $$ |e^{i\theta} -1| \leq |\theta| $$ I am studying the proof of differentiability of Fourier Series, and my book used this lemma. How does it work?
15
votes
5answers
2k views

Prove that the equation $x^{10000} + x^{100} - 1 = 0$ has a solution with $0 < x < 1$

Prove that the equation $x^{10000} + x^{100} - 1 = 0$ has a solution with $0 < x < 1$. This is a homework question. I know I could probably find a solution that would complete the proof, ...