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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28 views

Show that $f^{-1}$ is continuous

Let $E$ and $F$ two normed vector spaces, $A \subset E$ compact, $B \subset F$ and $f: A \to B$ is a bijective continuous function. As $f$ is bijective, we can defining the inverse function ...
0
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0answers
6 views

Determining the unit normal field of a paraboloid $P$, and integrating a vector field over $P$

Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are ...
-2
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1answer
47 views

Hard time on understanding real analysis. [on hold]

I am learning real analysis now but I really dislike the notion of limit, infinity... They seems to generate lots of paradoxes and unreasonable results. For example, when I am reading the uniform ...
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0answers
26 views

harmonious equations $y' = f(y/x)$

There is a wired Theorem that comes about when considering whether a function is harmonious (in the seance of the title definition). I was un-able to prove it, or to find a prof to it. Can any one ...
1
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2answers
29 views

Proving the convergnce of a sequence

So, I have to prove that the sequence defined as $a_{n+1}=\frac{6(1+a_n)}{7+a_n}$ converges and then find the limit. I have few questions; Do i have to assume that $a_n \geq 0$ or $a_n \leq0$. ...
1
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1answer
13 views

Length of the Union of Intervals is less than the Sum of Each Length of Intervals?

I am reading Royden and Fitzpatrick's book on Real Analysis, and I have a question about the length function on an interval in $\mathbb{R}$. Is it true that given $\{I_n \}_{n=1}^{\infty}$ of ...
0
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1answer
22 views

prove $\sup X \le \sup Y$ if $X$ is a subset of $Y$ both sets are nonempty and $Y$ is bounded above

Working through foundations of mathematical anlysis by johnsonbaugh per suggestion and wondering if the following proof works? (no solutions to book) Problem: Let X and Y be nonempty subsets of real ...
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0answers
17 views

Uniform continuity of this function

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. We assume that $f$ satisfies the following property: For every sequence of real numbers $(x_n)_n$, there exist a subsequence $(x_{\phi ...
0
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1answer
18 views

Proving measurability in $\mathbb{R}^2$

I am given the problem: suppose for measurable, real-valued functions $f$ and $g$, and an open set $A \subset \mathbb R ^2$, prove that $\{x \in \mathbb R : (f(x),g(x)) \in A\}$ is a measurable set. ...
0
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1answer
35 views

If $\sum_{n=1}^{\infty}x_n^2<\infty$ and $\sum_{m=1}^{\infty}x_n^2<\infty$, is $\sum_{k=1}^{\infty}(x_n)_k^2(x_m)_k^2<\infty$?

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ Prove that $l^2$ is complete with ...
4
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3answers
93 views

Limit of $a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$

Find a limit of sequence: $$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$ $$a_1=0,a_2=0$$ I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by ...
1
vote
1answer
12 views

What is a cluster point of this sequence?

I'm trying to make sure I have correctly understood Bolzano-Weierstrass, which states that every compact subset of $\mathbb{R}^{n}$ is sequentially compact, which means that if $A$ is a compact subset ...
0
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1answer
10 views

Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
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1answer
38 views

Help understanding theorem proof

So this is my first semester taking a Real Analysis class. We are using the book Introduction to Analysis by Gaughan 5th ed. This is my first real Math class and I'm really excited but I am having ...
1
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1answer
30 views

Function between two metric spaces?

I need to come up with: two metric spaces ( X , d ) and ( Y , p ) A continuous function f: X → Y A Cauchy sequence {xn} in X that isn't mapped to a Cauchy sequence in Y My idea was to make ...
1
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1answer
16 views

Proof a real functional is continuous in $C_{[a,b]}$ (verification)

I wish to have some feedback on the following proof of the claim below, either if it is correct, what to fix, or other suggestions. Claim: Let $\psi :[0,1] \times \mathbb{R} \to \mathbb{R}$ be a ...
2
votes
2answers
47 views

Prove that $\lim_{x \to +\infty} \frac{f(x)}{x} = L$ if $\lim_{x \to +\infty} [f(x+1) - f(x)] = L \space$

Let $f:[0, +\infty) \rightarrow \mathbb{R} $ be a bounded function in each bounded interval. If $$\lim_{x \to +\infty} [f(x+1) - f(x)] = L$$ then $$\lim_{x \to +\infty} \frac{f(x)}{x} = L$$ I tried ...
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2answers
15 views

Show that $Gr(f)$ is compact

Let $A \subset \mathbb{R}^n$ a compact and $f : A \to \mathbb{R}^m$ a continuous function. Let the graph of $f$ $$Gr(f) = \{(x,f(x) : x \in A)\}.$$ Show that $Gr(f)$ is compact. My proof : ...
0
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1answer
16 views

Definition of continuity up to the boundary

Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does it mean $f\in C(\bar{\Omega})$, i.e. what does it mean $f$ to be continuous at $x \in \partial \Omega$, maybe $$\forall \epsilon >0 ...
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0answers
30 views

How to relate the following function with Thomaes function

I am wondering how I can go about comparing the following functions $f(x)=\begin{cases} 1/n &\text{if $x=\frac{1}{n}$} \\ 0 &\text{else} \\ \end{cases}$ On the interval $[0,1]$, with the ...
0
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1answer
23 views

Uniform Convergence of Series tends to $f(x)$

The question is to prove that: $$\frac{f_1(x)+\cdots+f_n(x)}{n}$$ tends to $f(x)$ uniformly on $E$, as $n$ tends to infinity. I am not sure how to do this generically without an actual sequence. ...
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2answers
32 views

Does a bounded sequence in $C^k$ have a convergent subsequence in $C^{k-1}$

Let $K \subset \mathbb{R}^n$ be compact, let $k \in \mathbb{N}$. Let $\{f_n\} \subset C^k(K)$ be a bounded sequence w.r.t $C^k$ norm. Does it have a convergent subsequence in $C^{k-1}(K)$. If so how ...
0
votes
1answer
37 views

Splitting integral !!

i have this simple question that make me really confused : let $\phi$ a smooth function with compact support and $p>0,t\ge 0$ and : \begin{equation} U_p= \left\lbrace \begin{array}{ccc} 0 & ...
2
votes
2answers
56 views

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero?

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero? I was talking to a friend about it but neither of us were able to come up with anything. We're not sure if $\pi$ is essential ...
0
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2answers
63 views

Is there a nice open set proof that multiplication is continuous?

For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the ...
0
votes
1answer
41 views

Calculate the product limit by using elementary high school techniques

$$\lim_{n\to\infty} \prod_{i=2}^n(1-{1\over{i+1 \choose 2}}) $$ This is a problem i have encountered in one of my textbooks.Solve it by using high school methods for real analysis.
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2answers
22 views

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,…$

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,...$ I've tried to isolate n but its impossible to do.
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0answers
30 views

Calculus of integral over balls

I have this: $$\int_{B_{r\lambda_n/2}(y_n)\cap[B_{\lambda_n R}(0)\setminus B_{\lambda_n r}(0)]} x_1 |\nabla u_n|^p dx+\int_{B_{r\lambda_n/2}(y_n)\setminus[B_{\lambda_n R}(0)\setminus B_{\lambda_n ...
0
votes
1answer
59 views

Solve $x^2 = 2^x$. [duplicate]

One can see that the solutions are $x=2, 4$ and $x=-0.77$(approximately) seen from the graph. I am posting this to find if there is a way to solve this and find solutions like polynomial equations. ...
0
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1answer
33 views

If $f$ is differentible at a point $x \in [a,b]$, then $f$ is continuous at $x$.

Proof. As $t\rightarrow x$, we have, by Theorem 4.4, (Baby Rudin, p.104) $f\left( t\right) -f\left( x\right)$ = $\dfrac {f\left( t\right) -f\left( x\right) } {t-x}\cdot \left( t-x\right) \rightarrow ...
3
votes
2answers
28 views

Series of functions converge uniformly but sequence of functions does not

Given $a>1$ and $$f_{n}(x)=\frac{1}{1+n^{a}x^{4}}$$ I'm asked to show that for any $\delta >0$, the series of functions $\sum f_{n}(x) $ converges uniformly for $\{x \in \mathbb{R} | |x| \geq ...
1
vote
2answers
26 views

Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,…

I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a ...
0
votes
1answer
34 views

Prove that if $f:[a,+\infty [\longrightarrow \mathbb R$ is uniformly continuous, then $\lim_{x\to +\infty }f(x)=+\infty $ [on hold]

I have to show that if $f:[a,+\infty [\longrightarrow \mathbb R$ is uniformly continuous, then $\lim_{x\to +\infty }f(x)=+\infty $. I spend very much time on it, and I can't conclude. How can I find a ...
2
votes
3answers
99 views

Russell's paradox question

Tao's analysis book uses following example for Russell's paradox: $$P(x) \Longrightarrow `` x\text{ is a set, and }x \notin x"\\ \Omega := \{x : P(x)\text{ is true} \} = \{x : x\text{ is a set and }x ...
0
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3answers
20 views

Finding disc of convergence

Find the disc of convergence $$\sum_{n=0}^\infty z^{n^{3}}$$ I have applied the ratio test but I can not seem to come up with a conclusion.
0
votes
1answer
17 views

Bounded sequences without a convergent subsequence converging in a different metric

Given the metric $$d(x,y)=\text{min}\{1,|x-y|\}$$ on $\Bbb{R}$.There is a bounded sequence in $(\Bbb{R},d)$ without a convergent subsequence. Prove that a sequence in $(\Bbb{R},d)$ converges iff it ...
0
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0answers
41 views

Bound the first derivative of the following function: $f(x)=g(x)+h(x)$

Consider a decreasing function $f(x)$, resulting from the sum of two other decreasing functions: $f(x)=g(x)+h(x)$. All these 3 functions are positive. In addition, we have $g(0)=h(0)=1$. Further, we ...
1
vote
1answer
16 views

Find a power series centered at the origin that satisfies the Bessel

Find a power series centered at the origin that satisfies the Bessel differential equation $$zf''(z)+f'(z)+zf(z)=0$$ with initial condition $f(0)=1$. Show that this series converges for all z in C. I ...
1
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1answer
27 views

$A$ and $B$ be non-empty bounded set of real numbers, give a counter example to the following.

Assume $A \cap B \neq \emptyset$. Find a counter-example to the claim: $\sup(A \cap B) = \min\{\sup(A), \:\sup(B)\}$ I cant seem to find a counter example to the above claim, can anyone provide a ...
1
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2answers
48 views

How can I show that this function is discontinuous at the point $x=1$?

Suppose you had the function $$ f(x) = \; \text{ the integer part of } x $$ I wish to show that this is not continuous at the point $x=1$, which I will try to do by showing that $\lim_{x \rightarrow ...
0
votes
0answers
22 views

Find multiple roots of a three-dim system

Consider the three equations $$ y-x^2=0,\quad z+xy=0,\quad -y-z+x^2-xy+y^2+z^2-x^4=0. $$ How can I find multiple roots of this? Is it allowed to reduce the system as far as possible and then to find ...
0
votes
0answers
21 views

Parameterization which is closed under addition

Suppose $\beta_1(t)$ and $\beta_2(t)$ are two parametric curves defined on $[0,1]$. Let $\beta_1^*(t)$ and $\beta_2^*(t)$ are two re-parametrized of the above curves. Now, I looking for a ...
3
votes
1answer
54 views

How to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x+y)\le h(x)+h(y)$?

Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function and $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
0
votes
2answers
25 views

What values of $b$ such that $f_n(x)=b\cos\left(\frac{x}{n}\right)$ converges uniformly?

For what values of $b$ does the sequence of functions: for each $n\in\mathbb{N}$, let $$f_n(x)=b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1]$$ converge uniformly in the space $C[0,1]$ equipped ...
0
votes
2answers
55 views

A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
2
votes
2answers
79 views

Theorem 2.43 in Baby Rudin: How to understand the proof?

Here's Theorem 2.43 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $P$ be a non-empty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Here's the ...
4
votes
1answer
51 views

Check the proof of $||x||^2$ is not a norm

Show if $f$ is a norm: For $\mathbb{R}^n$, Define $f: \mathbb{R}^n \rightarrow \mathbb{R} $ by $ f(x) = \|x\|^2$ =$\sum_{n} x_n^2 $ I tried to solve if $f$ satisfies the three properties of a norm: ...
0
votes
1answer
30 views

Why does this follow from the triangle inequality?

Proving that differentiability implies continuity.
0
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2answers
28 views

Which metric is used in this limit

Rudin's Principles of Mathematical Analysis(3rd ed) says on page 53 that 'If $\{p_n\}$ is a sequence in $X$ and if $E_N$ consists of the points $p_N,p_{N+1},p_{N+2},\dots$, it is clear from two ...
-1
votes
1answer
46 views

Prove that a function is Riemann integrable directly, using $\epsilon-P$

I know there already are questions like these, but I still don't understand how to prove it. Question: Prove that $f$ is Riemann integrable on $[0,1]$ if $$f(x) = \begin{cases} x^2 \sin (1/x) ...