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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3
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4answers
430 views

Taking limits on each term in inequality invalid?

So this inequality came up in a proof I was going through. $$c - 1/n < f(x_n) \leq c$$ Where $c$ is a real number, $f(x_n)$ is the image sequence of some arbitrary sequence being passed through a ...
1
vote
1answer
41 views

Non-Cauchy products for series?

Let $\sum a_n$ and $\sum b_n$ be two absolutely convergent series. My question is Is it possible to take the product of the two series $(\sum a_n)(\sum b_n)$ and get a result different from the ...
6
votes
3answers
131 views

Is it true that $\sin x > \frac x{\sqrt {x^2+1}} , \forall x \in (0, \frac {\pi}2)$?

Is it true that $$\sin x > \dfrac x{\sqrt {x^2+1}} , \forall x \in \left(0, \dfrac {\pi}2\right)$$ (I tried differentiating , but it's not coming , please help)
2
votes
1answer
154 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
0
votes
0answers
22 views

for every $p\in f^{-1}(\{a\})$ there is an open neighbourhood $W\subset U$ such that $f^{-1}(\{a\})\cap W$ is the graph of a $C^1$-function

Let $U\subset\mathbb{R}^n$ open, $f:U\to\mathbb{R}$ continuously differentiable and $a\in\mathbb{R}$ such that $Df(p)\not= 0$ for all $p\in f^{-1}(\{a\})$. I want to know how to prove: for every $p\in ...
1
vote
1answer
82 views

A mean value theorem involving two functions [duplicate]

Let $f,g:[a,b] \rightarrow \mathbb{R}$ be continuous in $[a,b]$ and differentiable in $(a,b)$. Prove that there is a point $c \in (a,b)$ such that: $$[f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c).$$ I ...
3
votes
2answers
113 views

Prove for any set $E\subset R$ with lebesgue measure 1 there exists a subset with lebesgue measure 1/2.

Prove for any set $E\subset R$ with Lebesgue measure 1 there exists a subset with Lebesgue measure 1/2. It looks easy but I have tried for an hour and could not find a way to prove it. Can anyone ...
2
votes
3answers
253 views

A confusion in a calculation with complex numbers

Consider the followings: $$ 1+e^{ix}+e^{2ix}+e^{3ix}= \dfrac{1-e^{4ix}}{1-e^{ix}} $$ Then, we take absolute square to the both sides $$ |1+e^{ix}+e^{2ix}+e^{3ix}|^{2}= \dfrac{1-\cos4x}{1-\cos x} $$ ...
2
votes
1answer
37 views

Interchanging sum and differentiation, almost everywhere

Let $\{F_i\}$ be a sequence of nonnegative increasing real functions on $[a,b]$ with $a<b$ such that $F(x):=\displaystyle\sum_{i=1}^\infty F_i(x)<\infty$ for all $x\in [a,b],$ then show ...
0
votes
2answers
72 views

Uniform convergence of improper integrals

I'm having trouble w/ the following result, which is Ch 6, Thm 15, in Buck's Advanced Calculus: If $f(x,t)$ is continuous on $x\geq b$,$t\geq c$, $\int_c^\infty f(x,t)\;dt=F(x)$, uniformly on $x\geq ...
0
votes
4answers
71 views

Formal analysis proof for specific limit $\large{|x - \frac{p}{q}| < \frac{1}{n}}$ [closed]

If $x \in \mathbb{R}$ and $n \in \mathbb{N}$ then there exists $p, q \in \mathbb{Z}$ such that $$\left|x − \frac{p}{q}\right| < \frac{1}{n}.$$ Do we use the Archimedian Principle to prove ...
0
votes
1answer
41 views

Local Lipschitz continuity

In some proof I have seen the author use that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and bounded, then it is locally Lipschitz continuous. I have never seen that before and I don't find ...
11
votes
1answer
289 views

Every function $f: \mathbb{N} \to \mathbb{R}$ is continuous?

This is a question that came up as a true false question in my textbook, and I was wondering what you thought of my reasoning. I claim that even though a graph of such a function doesn't look ...
0
votes
1answer
866 views

The boundary of the union of two sets is a subset of the union of boundaries

I'm stuck on trying to get this proof started. I want to prove that $\delta(S_1 \cup S_2)\subset \delta S_1\cup\delta S_2$, where $S$ is some set. I don't need a full proof, just a hint to get ...
2
votes
1answer
84 views

Is there measurable function defined on unmeasurable set?

In my textbook, Lebesgue measurable function is defined as for every finite $a$, the set $\{x\in E:f(x)>a\}$ is a measurable set of $R^n$. And it further states $E=\{x\in ...
1
vote
1answer
67 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
0
votes
0answers
55 views

Books for Real analysis similar to or having same essence like Charles Pinter Abstract algebra

I am looking for book which is similar to spirit of pinter's book mentioned in question that is which is suitable for self study and beginners. Can anyone recommend? Thanks
0
votes
2answers
57 views

Is it true that a mapping between metric spaces is continuous iff the image of every open set is open?

Just want to change Rudin theorem 4.8 a bit and see if this works. The original theorem is ... $f$ is continuous iff $f^{-1}(V) $ is open in $X$ for every open set $V$ in $Y$. If I change the ...
0
votes
1answer
28 views

Statement about the gradient

Let $f \in \mathcal C^1(\mathbb R^n).$ If there exists $u \in S^{n-1}$ such that $$\nabla f(x) \cdot u \geq 0 \quad\forall x\in \mathbb R^n,$$ then $f(u) \geq f(0)$. How to prove this statement?
0
votes
2answers
29 views

Question on metric spaces.. 2 properties which I don't know whether they apply

Do these two properties hold in all metric spaces. In my textbook, it says they hold in spaces, that have defined scalar products, but I am interested if they hold in generally metric spaces: $$1.) ...
1
vote
0answers
38 views

Justification of surface area integral

I think that the Wikipedia article Surface Area says that a function $Area(S)$ defined on piecewise smooth surfaces which (i) agrees with the area of common surfaces, (ii) is additive over surfaces, ...
2
votes
3answers
69 views

Does $\int_a^\infty f$ exist iff $\int_a^\infty |f|$ exists?

My question is, does $\int_a^\infty f(x)dx$ exist if and only if $\int_a^\infty |f(x)|dx$ converges? Since $$\left|\int_a^\infty f(x)dx\right|\leq \int_a^\infty |f(x)|dx,$$ it's obvious that if ...
2
votes
1answer
48 views

Example where partial derivatives commute but are not continuous.

I am looking for an example of a function $f:\mathbb R^2\to\mathbb R$ such that there is a point $x\in\mathbb R^2$ with the following properties: 1) All partial derivatives of second order exist in a ...
3
votes
2answers
275 views

Show that $f(x) = x^2$ is not uniformly continuous on $[0,\infty)$

Ok, I know the same question has already been asked here, and I am not looking for an answer even though my proof looks kind of the same. But, I need to know whether or not I am on the right track. ...
3
votes
1answer
47 views

Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
0
votes
1answer
33 views

A difficult question about almost everywhere valid properties

Let $\mu$ be a measure and $[f]\in L^2(\mu)$, i.e. $$[f]=\left\{g\in\mathcal{L}^2(\mu):f\equiv g\;\;\;\mu\text{-almost everywhere}\right\}$$ Moreover, let $x^+:=\max(x,0)$ for $x\in\mathbb{R}$. ...
1
vote
0answers
51 views

The difference between a lemma and the Egorov's theorem

In my textbook, Egorov's theorem is proved with the help of a lemma. However, I have difficulty understanding the difference between the lemma and the theorem. The theorem is The lemma is ...
0
votes
1answer
240 views

Every step function is a linear combination of elementary step functions.

If $J$ is any subinterval of $[a, b]$ and if $\phi_J (x) := 1$ for $x \in J$ and $\phi_J (x) := 0$ elsewhere on $[a, b]$, we say that $\phi_J$ is an elementary step function on $[a, b]$. Then to ...
2
votes
2answers
59 views

Necessary condition for local maximum

Let $\Omega\subset \mathbb{R}^n$ open, bounded and let $f:\Omega\to\mathbb{R}$ be a $C^2$-function. I want to prove: Necessary for a interior maximum $x_0\in\Omega$ is that $D^2f(x_0)$ is negative ...
5
votes
3answers
161 views

Question about required rigour with intro real analysis text

I have just begun trying to self study introductory analysis and I am just having some questions about being specific on rigour. In the book I am using, titled Introduction to Real Analysis, 4th ...
0
votes
2answers
210 views

$A$ convex subset of a set has 'smaller' boundary than the set?

Let $A$ & $B$ be subsets of the real plane. Show that if $A$ is convex and is contained in $B$, which is a bounded set, then the length of the border of $A$ is $\leq$ than that of $B$.
6
votes
3answers
204 views

Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$

Use the mean value theorem to prove that if $0 \leq x \leq 1$, then $$x\sqrt{1-x^2} \leq \sin x \leq x$$ The theorem guarantees the existence of a point, but not an inequality, so I don't know how to ...
4
votes
2answers
61 views

Prove $f_n\to f'$ uniformly on $[0,1]$

Let $f:[0,2]\to \Bbb{R}$ be a continuously differentiable function. Let us define $f_n:[0,1]\to \Bbb{R}$ by $f_n(x)=n(f(x+{1 \over n})-f(x))$. Prove $f_n\to f'$ uniformly on $[0,1]$. I know that ...
2
votes
0answers
24 views

Can an equation be shown to be valid through logic over an continuous range?

I may be asking the impossible - but would appreciate it if someone else were to confirm this for me, rather than me just thinking this... I have a black box function, $f(x)$ that I don't know ...
0
votes
2answers
99 views

Show that $\sup (A\cdot B)=\max\{\sup A\cdot\sup B, \sup A\cdot\inf B,\inf A\cdot\sup B,\inf A\cdot\inf B\}$

Given nonempty subsets $A$ and $B$ of positive real numbers, define $$A\cdot B=\{z=x\cdot y:x\in A,\,y\in B \}$$ show that if $A$ and $B$ are bounded sets of real numbers, then $$\sup(A\cdot ...
3
votes
1answer
238 views

Total variation measure vs. total variation function

Let $a, b \in \mathbb{R}$ with $a < b$ and define the compact interval $I := [a, b]$. Let $g, h : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous on $I$ and constant on ...
0
votes
1answer
29 views

Polynomial inequalities of the form $C P_2 \leq P_1 \leq D P_2$

Let $P_1$ and $P_2$ be polynomials in $\mathbb{R} [x_1, \ldots, x_n]$ of the same degree. Under what conditions are there $C,D \in \mathbb{R}$ so that $C P_2 \leq P_1 \leq D P_2$ (as functions)? ...
4
votes
1answer
3k views

“Fat” Cantor Set

So the standard Cantor set has an outer measure equal to 0, but how can you construct a "fat" Cantor set with a positive outer measure? I was told that it is even possible to produce one with an outer ...
6
votes
4answers
1k views

Is the Cantor set a subset of rational numbers, and is it countable or uncountable?

In Chapter 2 of Rudin's Priniciples of Mathematical Analysis, Rudin takes the Cantor set as an example of a perfect set in $\mathbb{R}^1$ which contains no segment. Here's the construction of the ...
3
votes
2answers
96 views

Cantor Sets in perfect sets in the Real numbers

My thesis is related with the Cantor sets. I was reading a lot of papers, blogs, etc, in order to look for the mean properties of these sets. In one blog a read a proposition. ''Every perfect set ...
1
vote
3answers
78 views

False proof why $C^1$ implies locally Lipschitz

I produced a (possibly) false proof of why $C^1$ implies locally Lipschitz for $f: \mathbb R^n \to \mathbb R^n$. Please could someone tell me where my mistake is? Proof: Let $f: \mathbb R^n \to ...
1
vote
1answer
196 views

how to define that a nonlinear operator is bounded and continuous

We always see the definition of bounded and continuous linear operator. I am wondering how to define that a nonlinear operator is bounded and continuous. Is there any book providing this definition?
1
vote
1answer
121 views

Help understanding proof on Jensen's Inequality

I need help understanding the proof for Jensen's inequality in "Real and Complex Analysis" by Rudin. 3.3 Theorem (Jensen's Inequality) Let $\mu$ be a positive measure on a $\sigma$-algebra ...
3
votes
1answer
69 views

Sum of quotients

Assume $0<x_i\leq y<z$ for $i=1\ldots,n$. Is there an easy argument to show $$\frac{x_1}{y}+\sum_{i=1}^{n-1} \frac{x_{i+1}}{x_i}+\frac{z}{x_n}\geq n+\frac{z}{y}?$$ For $n=1$ the statement is ...
0
votes
4answers
154 views

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$. Should I look at this series as: $\sum_{n=1}^{\infty}({n!x^{(n-1)!})x^{n}}$? I am really confues here. In addition, any attempt to ...
0
votes
2answers
38 views

How to prove that E's limit point must be in E? (rudin)

How to prove that E's limit point must be in E? Thm 2.23 E's open iff E_c is closed. First, suppose E_c is closed. Choose x belongs to E. Then x doesn't belongs to E_c, and x is not a limit point ...
2
votes
1answer
97 views

Is the space of continuous functions with bounded variation separable?

Let $$BVC([0,1]) = \{f:\mathbb [0,1] \to \mathbb R,\, f\in BV([0,1])\cap C([0,1]),\, f(0)=0\},$$ and $$\|f\|_{BVC}=\mathrm{Var}(f).$$ Is $BVC([0,1])$ separable?
9
votes
1answer
448 views

Function that decays faster than any polynomial, but not in the Schwartz space?

Motivated by the very restrictive condition imposed in the definition of the Schwartz space, I was wondering about the following question. Is there a $C^\infty$ function that decays faster than ...
0
votes
1answer
80 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
0
votes
3answers
50 views

Calculate roots from $\frac{x \cosh(x) - \sinh(x)}{x^2}$

I want to solve the following equation $$f(x) = \frac{x \cosh(x) - \sinh(x)}{x^2} = 0$$ Because the term above is undefined for $x = 0$ I calcuted $$\lim_{x \rightarrow 0}\frac{x \cosh(x) - ...