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.

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8
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1answer
560 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
3
votes
3answers
112 views

To prove the sum is convergent [duplicate]

Let$$a_n \ge 0$$ for all $n \in\Bbb N$. Show that if $$\sum_{n=1}^\infty a_n$$ converges, then $$\sum_{n=1}^\infty {\sqrt a_n\over n}$$ converges, too. The hint is to expand $$\left(\sqrt a_n-{1\over ...
0
votes
1answer
41 views

Is my proof about $\liminf$ of bounded real sequences correct?

Yesterday I asked in the question at A proof about the limit infimum of a bounded sequence about the proof of the following statement: Let $x_n$ be a bounded sequence of real numbers. Then the ...
3
votes
0answers
142 views

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
0
votes
1answer
41 views

How do the inner products on $L^2$ look like?

I was wondering whether all scalar products in $(L^2[0,1],\lambda)$ are given by $\langle f,g \rangle := \int f(x)g(x) \cdot w(x) d\lambda(x)$? If this is true, what are the exact conditions that we ...
3
votes
2answers
84 views

Let $a_{n+1}=cos(a_n)$ for $n\ge0$ and $a_0 \in [-\pi/2,\pi/2]$ Find $\lim_{n \to \infty}a_n$ if it exists.

Let $a_{n+1}=\cos(a_n)$ for $n\ge0$ and $a_0 \in [0,\pi/2]$ Find $\lim_{n \to \infty}a_n$ if it exists. I drew some sketches and it does seem like the limit exists, it's probably $x$ such that ...
2
votes
1answer
879 views

what prerequisite classes must I have before I take Abstract Algebra and Real Analysis at the undergrad level?

what prerequisite classes must I have before I take Abstract Algebra and Real Analysis at the undergrad level? Will these prerequisites help me with these two coursea at the graduate level? or do I ...
2
votes
1answer
71 views

The definition of Compactness for “set” and “space”

Compactness for "set" and "space" I was wondering if there is any significance between the two settings. Do we treat them as two different things? For example, let $(X,d)$ be a metric space with the ...
0
votes
1answer
41 views

Terminology: Alternatives for zero crossing

Is it correct to name the red and blue points hinge points, as an alternative to zero crossing? Or are their better terms to describe these points? Update I have several functions like these. I ...
76
votes
3answers
3k views

Is there an integral that proves $\pi > 333/106$?

The following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves ...
0
votes
1answer
55 views

Lower semicontinuity

Let $\Omega\subset\mathbb R^n$ be open and bounded. I consider a sequence $u_k:\Omega\to\mathbb R$ of smooth functions which converges uniformly to a function $u:\Omega\to\mathbb R$. Moreover, the ...
2
votes
3answers
59 views

Considering $\epsilon$ intuitively in limit proof

I'm having rather difficult time in trying to use $\epsilon$ argument appropriately. For example here is my simple $\epsilon$ proof in one question. The question is as follow: Prove if $s_n \geq 0$ ...
2
votes
1answer
55 views

approximation of $\log(1+z)=z$ as $z\to 0$

This is new to me and I have not done any asymptotic approximation. I don't understand how they get that $\frac{n}{N}$ stays close to $\frac{2}{3}$ as N goes to infinity. Also how do they do get ...
1
vote
1answer
28 views

Suppose $\{e_n\}_{n=1}^\infty $ is the orthonormal system of functions in $L^2[0,1]$, if $e_n \in C^1[0,1]$, then $e_n'(x)$ can be unbounded.

Suppose $\{e_n\}_{n=1}^\infty $ is the orthonormal system of functions in $L^2[0,1]$, if $e_n \in C^1[0,1]$ for all $n\in \mathbb{N}$, then $\sup_n \max_{x\in [0,1]}|e_n'(x)|=\infty$. I consider the ...
34
votes
3answers
586 views

When does a real function together with its derivatives form a basis?

Consider real functions defined on an interval, let's say $[-\pi,\pi]$. Let $s(x)$ be an analytic, periodic function such that all of its Fourier components are non-zero. We build a set of functions ...
1
vote
1answer
19 views

If a continuous function $f$ satisfying $D^+ f(x) \geq a>0, \forall x\in \mathbb{R}$, where $ a$ is a const, then $f$ is monotone?

If a continuous function $f$ satisfying $D^+ f(x) \geq a>0,\forall x\in \mathbb{R}$, where $ a$ is a const, then $f$ is monotone? Here $D^+ f(x)$ means the Dini's differential, $D^+ ...
0
votes
1answer
56 views

A Set That Is “Precisely” Measure-Dense [duplicate]

This question asks for a set of real numbers that is measure-dense, whose complement is also measure dense. In terms of $[0,1]$, the question asks for an $S$ such that for every open interval $I$ we ...
91
votes
9answers
6k views

Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^t dt$ is ...
8
votes
1answer
122 views

$\forall x \,\exists k$ s.t. $f^{(k)}(x)=0$, then $f$ is a polynomial

My friend sent me the following problem: Suppose that $f$ is real analytic on $(a,b)$, and that for all $x$ in $(a,b)$ there exists a non-negative integer $k$ such that $f^{(k)}(x)=0$. Show ...
1
vote
0answers
51 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
0
votes
1answer
41 views

Every path $f:[a,b] \rightarrow \mathbb {R}^n$ of class $C^1$ is rectifiable, and the lenght $\mathcal {l}$$(f)= \int_{a}^{b} |f'(t)|dt$

Good evening everyone! I am in trouble to demonstrate the folowing theorem: "Every path $f:[a,b] \rightarrow \mathbb {R}^n$ of class $C^1$ is rectifiable, and the length $\mathcal {l}$$(f)= ...
2
votes
1answer
62 views

Showing that a set is not infinite in measure

Suppose $f_n \geq 0$ for all $n \geq 1$, $f_n \to f$ a.e. on $[0, \infty)$ and there exists a constant $M>0$ such that $$ \sup\limits_{n} \int_{E} f_n(x)dx \leq M \mu(E)$$ for each measurable ...
1
vote
1answer
42 views

Composition and Limits

Suppose that $f$ is a continuously differentiable function with $\lim_{x \rightarrow \infty} f(x)=k$ and $g$ is a Lipschitz continuous function. Prove that $\lim_{x \rightarrow \infty} ...
0
votes
0answers
144 views

How do you use R to find the box counting dimension of a two dimensional set of data, or scatter plot?

I'm using the software R to do some analysis on some data sets for a graduate project. R has a package called "fractaldim", in this package is a function for finding the box counting dimension. The ...
5
votes
2answers
202 views

What is an advatage of defining $\mathbb{C}$ as a set containing $\mathbb{R}$?

It is a theorem that every field with least upper bound property and Archimedean property is isomorphic to each other. So it seems not necessary to define $\mathbb{R}$ exactly and we simply denote ...
32
votes
5answers
865 views
0
votes
1answer
50 views

Distance Between Points and Sets

You are given two sets $A, B \subset \mathbb{R}^n$. Show that $\{ x \mid d(x, A) < d(x, B) \}$ is open.
20
votes
4answers
976 views

Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only

I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in ...
3
votes
1answer
83 views

How to show a set is compact in a function space?

I have a question asking if $\{f_n\}$ is a compact in $C_b([0,\infty))$ (bounded continuous) with $||\cdot||_{L^\infty}$. The sequence is $$f_n (t) = \sin\sqrt{t+(2n\pi)^2},$$ I have showed that ...
2
votes
1answer
122 views

A continuous function that attains neither its minimum nor its maximum at any open interval is monotone

Let $f: \mathbb R\to \mathbb R$ be a continuous function such that $f$ attains neither its minimum nor its maximum at any open interval $I \subseteq \mathbb R$ , then how to prove that $f$ is ...
5
votes
1answer
79 views

If $f: A\to\mathbb R$ one-to-one but not monotone, there exist $x,y,z\in A$ with $x<y<z$ such that $f(x) < f(y)$ and $f(y) > f(z)$ (wlog)

The following result is part of the folklore, but I'd like to have a standard reference for something that I am writing: If $A \subseteq \mathbb R$ and $f: A \to \mathbb R$ is one-to-one but not ...
1
vote
2answers
50 views

Fixed points of a certain type of functions with intermediate value property

Let $f: \mathbb R\to \mathbb R$ be a function, having intermediate value property, such that $f(f(x))=x , \forall x \in \mathbb R$, then is it true that either the set of fixed points of $f$ is ...
1
vote
1answer
23 views

Showing that a function is bounded in $L^1$ given a bound on its distribution function

Let $f \in L^2((0,T)\times\Omega)$ where $\Omega$ is a compact manifold. Suppose I know that for every $k > 0$, $$\mu(\{|f| > k\}) \leq Mk^{-\frac 12}$$ for some constant $M$ (which is ...
1
vote
1answer
125 views

A proof about the limit infimum of a bounded sequence

I tried to find a proof for this statement: If we have a bounded sequence $x_n$, then the limit infimum is defined as $a=\liminf_n x_n$ such that $a$ is the largest of real numbers which have the ...
0
votes
1answer
216 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...
1
vote
1answer
53 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...
1
vote
0answers
24 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
1
vote
1answer
151 views

Is my proof on showing that the set of monotone functions on $[a,b]$ has cardinality of continum correct?

I was given an exercise problem to show that the cardinality of the set of all monotone functions on $[a,b]$ is $\aleph$. I came out with a proof which I am not sure if it is correct. My proof: Let ...
0
votes
0answers
59 views

Prove that the function $g(·)$ is twice continuously differentiable and that $g′′(α) ≥ 0$ for all $α ∈ \mathbb{R}$, i.e.

Let $f$ be a real Lebesgue measurable function on the interval $[0, 1]$ such that $∥f∥∞ < ∞.$ For $α ∈ \mathbb{R}$ define a function $g(α)$ by $g(α) = \log \int_0^1\exp[αf(x)] dx .$ (a) Prove that ...
2
votes
4answers
68 views

Determined or not?

the function $\dfrac {2x}{3x-\sqrt{x} }$ is not derterined for values of $x$ equale or samller than zero, though when I take the limit $ \lim_{x \to 0^+} \dfrac {2x}{3x-\sqrt{x} }$ the output is zero ...
2
votes
4answers
89 views

How to find a function $\phi(x)$ such that $\sqrt{1+y^2} - \sqrt{1+x^2} \geq \phi(x) (y-x)$ for each $x,y\in \mathbb{R}$

How to find a function $\phi(x)$ such that $\sqrt{1+y^2} - \sqrt{1+x^2} \geq \phi(x) (y-x)$ for each $x,y\in \mathbb{R}$. Here are some of my ideas: Also by applying Mean Value theorem, we know that ...
1
vote
1answer
142 views

Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
1
vote
1answer
30 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
3
votes
1answer
65 views

There exist $c(c\in \left(\frac{1}{2},1\right)$ s.t $2 \int _{0}^{1}f(x)dx=\frac{f(c)}{c} $

I would appreciate if somebody could help me with the following problem: Let $f(0)=0,f(1)=1,f'(x)>0,f''(x)<0$ on $[0,1]$ Q: Under the proposition is true of false? There exist $c(c\in ...
0
votes
1answer
57 views

Do all series have a closed form representation of their partial sum? If not, can we feasibly prove that this is not the case?

The question was motivated by the way in which we approach the convergence and divergence of some series. During my undergraduate analysis course one of the only times in which the partial sum was ...
2
votes
2answers
67 views

Compute limit of a function

Compute: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$ WolframAlpha tells me it's 1/6. Any nice idea how to rewrite that expression? Thanks!
-1
votes
2answers
80 views

a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [closed]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
2
votes
2answers
95 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
0
votes
1answer
56 views

Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...
1
vote
1answer
226 views

Does every inner product space has an orthogonal basis?

It is proved that every inner product space has a basis $W$, but I am not sure if every inner product space has an orthogonal basis? It is known that every inner space has a maximal orthogonal set ...