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, and integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

1
vote
3answers
29 views

An exercise on compactness on $L^1$

If $g$ is a nonnegative measurable function on $[0,1]$, and let $K=\{f\in L^1([0,1]) :|f|\leq g \,\,\,\, a.e.\}$, prove the following: (a)$K$ is closed, (b)If $K$ is compact then $g\in L^1$. Part ...
3
votes
4answers
41 views

For any closed subset of $\mathbb{R}$ there is a sequence in $\mathbb{R}$ whose sequential limits is equal to the that subset

Question: Let $A$ be a closed subset in $\mathbb{R}$. Prove that there exists a sequence $x_n$ in $\mathbb{R}$ whose set of subsequential limits is exactly equal to $A$. My approach: I think this ...
2
votes
0answers
30 views

Help with a proof regarding simple functions.

The question is If $f≥g≥0$, then there exists non-negative measurable simple functions $f_k↗f$ st. $f_k≥g$ for all $k$. My attempt. Using a theorem in my text book For every non-negative ...
6
votes
2answers
79 views

Condition to guarantee $f=0$ on $[a,b]$

I have been stuck for several days on this old Analysis problem (I am doing some study on my own). I have tried several things (which I'll indicate below), but I cannot seem to figure it out. Here ...
0
votes
2answers
51 views

Is the Commutative Property of Addition on the Reals a Postulate or Proven?

My Calculus book from back in the day (Calculus Second Edition Michael Spivak) starts out by stating 12 basic properties of numbers which he labels P1-P12. He states: "Most of this chapter has ...
0
votes
0answers
20 views

Maximal cake-cutting

Alice and George divide a cake between them. The cake is a 1-dimensional interval and both players value the entire cake as 1. The valuations of the players are represented by non-atomic measures on ...
0
votes
1answer
42 views

The Real Numbers and Real Analysis

Prove that $\sqrt{1+4x}<2x+1$ for all $x\in(0,\infty)$ using the Intermediate Value Theorem of Derivatives.
2
votes
1answer
41 views

In which sense does Cauchy-Riemann equations link complex- and real analysis?

On page 12 of Stein, Shakarchi textbook 'Complex analysis', the authors state that the Cauchy-Riemann equations link complex and real analysis. I have completed courses on real and complex analysis, ...
0
votes
3answers
19 views

Fixpoints and continuity

I don't understand why this is true: If $f:[0,1]\rightarrow[0,2]$ is a continuous function then exists $x \in [0,1]$ such that $f(x)=2x$ I don't understand why such a point exist. Why is there not ...
0
votes
0answers
33 views

Need help solving min, max, inf and sup of sequence!

We've been given the sequence $x_n=(-1)^n \cdot \frac{\sqrt{n}}{n+1}+\sin \frac{n \pi}{2}$. I have to find the min, max, inf and sup (if they exist), and also find the points of accumulation. Any ...
1
vote
0answers
19 views

Hausdorff and surface measure

I know that Lebesgue measure in $\mathbb{R}^n$ coincides with Hausdorff measure $\mathcal{H}^n$. But I'd like to see the proof that also surface measure on manifold $M$ coincides with the Hausdorff ...
1
vote
0answers
20 views

Limit of the “oscillation” of a function

Let $f:[a,\infty) \rightarrow \mathbb R$ be a bounded function, and for each $t\geq a$ the functions $M_t = \sup f|_{I}$, $m_t = \inf f|_{I}$ where $I=[t,\infty)$. Define the function $\omega_t = M_t ...
3
votes
2answers
64 views

How to show $\int_{[0, +\infty)} \frac{2}{1+x^2} dx$ Lebesgue integrable?

Definition of Lebesgue integral of simple function: We say that a simple function $\psi$ is Lebesgue integrable if the set $\{\psi \ne 0\}$ has finite measure. In this case, we may write the ...
0
votes
1answer
27 views

How to know which notion of convergence to use when proving density of a subspace

My question might be a little vague, but is there a way to know which type of convergence (i.e pointwise, uniform) to use when proving that a subspace is dense in a certain space. For example if we ...
4
votes
4answers
82 views

On extended real line, is $(-\infty,+\infty)$ still a closed set?

On real line $(-\infty,+\infty)$ is open as well as closed. On extended real line $[-\infty,+\infty]$, is $(-\infty,+\infty)$ still a closed set? Thank you.
6
votes
1answer
54 views

Closed form of series involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number and $\phi$ be the golden ratio, that $\phi = \frac{1+\sqrt{5}}{2}$. Find a closed form for the sum: $$\sum_{n=0}^{\infty} \frac{1}{(5\phi)^n(n+2)} ...
6
votes
1answer
121 views

Finding a point with $f(x)=f'(x)$

Let $f :[0,1] \rightarrow \mathbb{R}$ be a fixed continuous function that is differentiable on $(0,1)$ and such that $f(0)=f(1)=0$. Does there exist a $x_0 \in (0,1)$ such that ...
-1
votes
1answer
23 views

Another question on finiding special kind of power series [on hold]

Let $\sum a_nx^n$ be a real power series with finite positive radius of convergence $R$ ; then does there exist a non-constant real sequence $\{b_n\}$ such that $\sum b_nx^n$ is convergent for at ...
0
votes
0answers
38 views

Theorem 12.13 of Apostol's Mathematical Analysis (2nd Ed)

I had a little difficulty understanding Theorem 12.13 of Apostol's Mathematical Analysis, and was wondering if someone who had read this book could give me some pointers. Specifically, on page 360, ...
2
votes
1answer
38 views

Sequence equivalence [on hold]

We call two sequences are equivalent if $\lim_{n\rightarrow\infty}u_n/v_n=1$. (1) Using $e=\sum_{k} 1/k!$, find a simple, nontrivial (meaning not the same $u_n$) sequence equivalent to ...
4
votes
1answer
30 views

On finding special kinds of power series

Let $\sum a_n x^n$ be a real power series with finite positive radius of convergence $R$, then is it true that for every real number $s>0$ , we can find a real sequence $\{b_n\}$ (depending on $s$, ...
2
votes
3answers
32 views

If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0$ always? [duplicate]

If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0?$ My attempt: Suppose $dist~(E,F)=0.$ Then $\exists~e \in E,f \in F$ such that $~\forall ...
1
vote
1answer
17 views

Logarithmically bounded function fulfills $f(n) \le \lceil m \cdot \log_b r \rceil$ for certain numbers $n,m,r$

Let $f : \mathbb N \to \mathbb N$ be a function such that $f(n) \le 1 + \log_b n$ for some base $b$ and all $n$. Now let $n \in \mathbb N$ have the property that $$ \frac{r^m - 1}{r-1} \le n < ...
2
votes
1answer
23 views

An inequality involving supremum and integral

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ ...
2
votes
2answers
62 views

Is $\int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} dx dy dz$ finite?

My question is in the title : How could I prove that $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} \ \text{d}z \ \text{d}y \ \text{d}x $$ is finite (if it is) ? Thank you by ...
2
votes
1answer
24 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that ...
0
votes
1answer
21 views

Does almost everywhere differentiablty imply existence of weak derivitive?

Does almost everywhere differentiablty imply existence of weak derivitive? What about the converse? If not in general maybe on compacts?
1
vote
1answer
28 views

Approximation, $C^{1}$ function

I have a question about approximation by $C^{1}$ fuctions. Let $f:[0,1] \to \mathbb{R}$ be a Lipschitz continuous function. Question Let $\epsilon>0$. Can we find a $C^{1}$ fucntion $G$ ...
4
votes
2answers
50 views

Existence of a solution for a nonlinear ODE on $[0,\infty)$

I'd like to prove that the solution to the following IVP exists on $[0,\infty)$. The IVP is given by $$ \begin{cases} y'(t) = y^2 \cos(t)-ye^t \\ y(0)= y_0 \end{cases} $$ where $y_0 ...
1
vote
1answer
68 views

quantum mechanics violate Bell's inequality

I have this function $$ \begin{aligned} F\big(\theta_a,\theta_b,\phi_a,\phi_b\big) = \ & – \big[\cos \theta_a \cos \theta_b \big] – \big[\sin\theta_a \sin\theta_b \sin\phi_a \sin\phi_b\big] \\ ...
0
votes
2answers
59 views

Compute the integral over the volume of a torus,

In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., $$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$ Let $\Omega$ ...
2
votes
2answers
77 views

Convergence of Sequence

If I know that the sequence $\{a_n\}$ converges to $a$, then to prove that the sequence $\{ca_n\}$ (for a constant $c$) converges to $ca$, I would basically want $|ca_n - ca| < \epsilon$... So, to ...
1
vote
2answers
62 views

Historical Approach to $\lim_{x \to 0} \frac{e^{\alpha x} - e^{\beta x}}{x}$, without L'Hospital's Rule

I encountered this problem, amongst others, in the slightly older Calculus textbook Piskunov's Differential and Integral Calculus when I was working with a student: Calculate the limit $$ \lim_{x \to ...
1
vote
0answers
32 views

Can someone suggest some reference for a particular kind of infinite series

Can someone suggest some reference for the properties of the series $\lim_{n \to \infty} \sum_{i = 0}^{n} f(n, i)$ For example, when can we apply approximations to $f(n, i)$ that only works when $n ...
5
votes
3answers
225 views

vercongent sequences

Definition- We say a sequence $(x_n)$ verconges to $x$ if there exist an $\epsilon>0$ such that for all $N\in \Bbb{N}$, $n\ge N \implies |x_n-x|<\epsilon$. Loosely speaking, by convergent ...
1
vote
0answers
27 views

Measurability of functions with respect to $F(x,y) =xy.$

We know $F$ maps $[0,1]^2$ into $[0,1]$ and this induces a $\sigma$-algebra $\mathcal{M}=\{F^{-1}(B): B \text{ is Borel subset of }[0,1] \}$ on $[0,1]^2$. How can we describe the ...
0
votes
1answer
20 views

Prove that regular curves are locally invertible

Consider the function $F = (F_1, F_2)$ from $I = (a, b) \subset \mathbb{R}$ to $\mathbb{R}^n$ (without loss of generality, assume $n = 2$). Suppose $F$ is differentiable (i.e $F_1' = f_1$ and $F_2' = ...
3
votes
3answers
119 views

Undetermined vs. Undefined [duplicate]

This often comes up in precalculus and calculus, that is sometimes an expression will be said to undefined while at other times undetermined. What is the fundamental difference between the two? For ...
1
vote
1answer
20 views

A question about fixed-point iteration sequence of a two times continuously differentiable function

I am stuck at this problem: Let $g:[a,b]\to[a,b]$ be a 2 times continuously differentiable function that satisfy: for all $x\in [a,b]$, $g''(x)\neq 0$ And let $s$ be an arbitrary fixed-point of ...
1
vote
0answers
33 views

The differentiability of a function follows from the differentiability of its composition with others?

Let $\varphi:U \to V$, where $U$ and $V$ are open subsets of $\mathbb R^n$ and $\mathbb R^m$ respectively. Suppose that for every differentiable function $f: V \to \mathbb R$ function $f \circ ...
3
votes
2answers
34 views

How to show the inequality is strict?

This is an exercise from Rudin's Principles of Mathematical Analysis, Chapter $6$. Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) = 0$, and $$\int_a^b ...
1
vote
2answers
27 views

Show that $\exists q\in\mathbb{Q}:d-c>|q-\sqrt{2}|$ where $d>c\in\mathbb{R}$

I'm trying to show that if $d-c>0$, then $\exists q\in\mathbb{Q}:d-c>|q-\sqrt{2}|$. In the case where $d-c>\sqrt{2}$, we have: $$ \exists q\in\mathbb{Q}:\sqrt{2}>q>0 \implies ...
10
votes
1answer
143 views

Could it possibly have a nice closed form? $\int _0^1\int _0^1\frac{x y}{(x+1) (y+1) \log (x y)}\ dx \ dy$

Using multiple integrals it's not hard to show that the present integral reduces to some integral over squared digamma functions, but then things become harder. How would you tackle the problem? ...
4
votes
1answer
38 views

Continuity and Supremum

Attempt: I can't seem to gauge the points of continuity, for the attain supremum parts, I know I need to use the fact a continuous function on a closed, bounded interval is bounded and attains its ...
1
vote
2answers
71 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
5
votes
2answers
87 views

Problem 7 IMC 2015 - Integral and Limit

I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
3
votes
5answers
84 views

Elegant solution for $\int {\frac{\cos(y)}{\sin^2(y)+\sin(y)-6}}dy$

I have the following integral: $\int {\frac{\cos(y)}{\sin^2(y)+\sin(y)-6}}dy$ I already know the solution, but it needs three substitutions. Is there a simpler, more elegant way to go about this?
0
votes
0answers
16 views

$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n)$

Do you have a reference for the following intuitive result? Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequence of reals. Then $$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n).$$
0
votes
0answers
30 views

Proof that the sum of a certain infinite series can be bounded to zero

$\forall 0 < \alpha < 1$, there exists $\lambda > 0$, $k > 0$, s.t. $$ \lim_{n \to \infty} \sum_{w = 1}^{\lambda n} \binom{n}{w} \frac{1}{2^{\alpha n}}\left(1 +\left(1 - ...
1
vote
3answers
70 views

Does l'Hopital rule work for -inf/inf?

If you have an indeterminate form: $\frac{-\infty}\infty$ $\frac\infty{-\infty}$ $\frac{-\infty}{-\infty}$ does l'Hôpital's rule also apply?