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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27 views

Prove that the solution for $y'=y^3(1-\tan^2(\arcsin(y)))$ , $y(0)= {\pi \over 8}$ , is bounded.

I got this problem to prove, and I assume I need to use the existence and uniqueness theorem for non-linear ODE's, so I set $y' = f(x,y)$ and differentiating in respect to $y$ gives: $f_y(x,y)$. And ...
1
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1answer
14 views

Differentiable any finite number of times

Does there exist a pathological function which is differentiable any finite number of times as one wishes, but is not differentiable an infinite amount of times? Is it reasonable for such function to ...
0
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0answers
10 views

A faithful positive Radon measure

Let $X$ be a locally comapct and Hausdorff space. We say a positive Radon Measure on $X$ is faithful if $$0\leq f ~~~,~~~\int fd\mu=0\rightarrow f(x)=0 ~~\forall x\in X$$ Q: True or false: If there ...
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0answers
14 views

Support of a Radon measure

Let $X$ be a locally compact and Hausdorff space. For a given Radon measure $\mu$ on $X$, the support of $\mu$ is the smallest closed subset of $X$ with $|\mu|(X)=\lVert\mu\rVert$ (where $|\mu|$ is ...
0
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0answers
5 views

Nonlinear operator sends bounded set to relatively compact set

Consider $g$ a continuous function on $[a,b]\times\mathbb{R}$, and let $z_0\in\mathbb{R}$. Define the (nonlinear) operator on $C[a,b]$: $$Mv(x)=z_0+\int_a^x g(t,v(t))\,dt$$ for $x\in[a,b]$. Prove ...
0
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2answers
16 views

Lp space Inclusion Examples

I proved for a bounded set $\Omega$ and $1 \leq p \leq q \leq \infty$ that $L^{q}(\Omega) \subset L^{p}(\Omega)$. What is an example that would show strict inclusion, $ p<q$, and false if $\Omega$ ...
2
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0answers
16 views

Poincare Inequality in n-dimensions

I am trying to prove the Poincare Inequality on a n-dimensional box. That is a domain $ \Omega = (0,1)^n$ for $f(x) \epsilon H^{1}_{0}(\Omega) $, show there exists a constant $C$ such that ...
1
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1answer
30 views

Functions between metric spaces (and how they relate to closures of sets)

Let $(X,d)$ and $(Y , p)$ be metric spaces. Prove that if $f : X \to Y$ is continuous, then for any set $A\subset X$ with closure $\overline{A}$ we have $f(\overline{A})\subset \overline{ f(A) }$ ...
2
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1answer
20 views

A subspace of a mapping space?

We have a set $$ M=\{f:\mathbb{R} \rightarrow \mathbb{R}\mid f(1)>0\}\;.$$ I have never encountered this kind of set before. I assume it is correct to say that $M$ is a subspace of a mapping ...
0
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1answer
25 views

Why is continuity needed to substitute value of derivative inside Riemann-Stieltjes Integral?

Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ $$ \beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]} $$ Why is the additional assumption $f$ continuous and what ...
0
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1answer
30 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
3
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1answer
33 views

Determining an upper bound

I have a function $$f(\lambda)=n\ln(1-p+pe^{\frac{\lambda}{n}})-\lambda p$$ I need to prove that $$f(\lambda)\leq \frac{\lambda^2}{8n}$$ using Taylor expansion. I have used the taylor expansion for ...
0
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0answers
20 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
1
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1answer
36 views

Limit using definition

I am trying to find $$\lim\limits_{x \to \infty} n-\sqrt{n^2-4n} $$ using the definition of a limit. I have tried to multiply top and bottom by $n+\sqrt{n^2-4n} $ giving $\frac{4n}{n+\sqrt{n^2-4n}} ...
0
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2answers
22 views

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n\}$ is closed

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n \}$ is closed. I had to show it is compact, and I am done showing it is relatively compact, but now I am stuck showing it is closed. ...
0
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2answers
41 views

Show that $f$ is uniformly continuous - $\lim_{\|x\| \to \infty}f(x) = c$

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a continuous function. We suppose that there exist $c \in \mathbb{R}^m$ such that $$\lim_{\|x\| \to \infty} f(x) = c.$$ Show that $f$ is uniformly ...
0
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2answers
25 views

Compact set of a set $K$

Show that $K := \{x \in C[0,1] : x(0) \in [-3,4], |x(t)-x(s)| \leq d |t^2-s^2|, \forall t,s \in C[0,1]\}$ is compact. I already know that $C[0,1]$ is a compact set. So it is only necessairy to show ...
0
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0answers
16 views

What kind of information does the derivative of a sequence of functions give us?

When we derivate a function we know for example when then function increases and drecreases but when I derivate a sequence of functions I don't how to interpret the derivative since it is a collection ...
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2answers
32 views

Disc of convergence of a power series

Find the disc of convergence: $$\sum_{n=3}^\infty \left(1-\frac{1}{n^2}\right)^{-n^3}z^n$$ I have been manipulating the power series and I am pretty sure it has something to do with $e$ but I cannot ...
0
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2answers
33 views

What is the value of integral? [on hold]

Let $y(t)$ be a continuous function on $[0,\infty)$. If $$ y(t)= t\left(1-4 \int^t_0 y(x) dx\right) +4 \int^t_0 xy(x) dx$$ then what is the value of $\int^{\frac{\pi}{2}}_0 y(t) dt\,$?
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0answers
16 views

Decomposition of function of bounded variation

Suppose we have $f:\mathbb{R} \rightarrow \mathbb{R}$ which is of bounded variation. I would like to show that it can be presented as a sum of left and right continuous functions of bounded variation. ...
0
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2answers
22 views

Show if one series converges absolutely then so too does the other.

Task at hand: Let $a_n$ and $b_n$ be nonzero complex numbers for $n=1,2,3...$ . Suppose $\lim_{n\to \infty} \left|\frac{a_n}{b_n}\right|=l$ exists, and $l\neq0,\infty.$ show that if one of the series ...
1
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4answers
27 views

Pointwise convergence to zero, with integrals converging to a nonzero value

For $n\in{\mathbb{N}}$ let $$f_n(x)=nx(1-x^2)^n\qquad(0\le x\le 1).$$ Show that $\{f_n\}_{n=1}^\infty$ converges pointwise to $0$ on $[0,1]$. Show that $\{\int_0^1f_n\}_{n=1}^\infty$ converges to ...
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2answers
21 views

Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...
1
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1answer
72 views

Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

Having $f(z)=\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n$ I had to find what $\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n\sum\limits_{n=0}^{+\infty} \frac{D_n}{n!}z^n=\sum\limits_{n=0}^{+\infty} ...
1
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1answer
12 views

Heine definition of limit of a function at infinity using sequences

I couldn't find the answer neither on Google, nor this website, so decided to ask. The Heine definition of limit: from Wikipedia $\lim_{x\to a}f(x)=L$ if and only if for all sequences $x_n$ (with ...
0
votes
1answer
39 views

der(der(A)) and der(A)

I have a question about cluster points that would like to ask you, this question is one of the exercises in my textbook. Question: If $A$ is any subset of $R^d$, then $der(der($A$))$ is the set of ...
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0answers
20 views

Showing that the inferior integrals of two functions are equal.

Let $g:[a,b]\to\mathbb{R}$ Riemann integrable and put $h(x)=g(x)$ if $x$ is rational and $h(x)=g(x)+1$ if $x$ is irrational. Show that the inferior (Darboux) integral of $h$ is equal to the integral ...
1
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3answers
55 views

Solve the following sequence problem

Let a sequence be defined as $$a_n=\lim_ {x \to 0}{1-\cos (x)\cos(2x).....\cos (nx)\over x^2}$$ a)prove that the given sequence is monotonic and that it is not bounded above. b)calculate $$\lim_{n \to ...
0
votes
1answer
23 views

Bound the variation of this decreasing function

Let $f(x)$ be a decreasing function defined over the interval $[0,a]$, with $f(0)=b$. The first derivative of $f(x)$, which is negative, is such that $f^\prime(x) > g(x)$, or equivalently ...
1
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1answer
32 views

Show that $f$ is uniformly continuous - Compactness

Let $K := \{x \in C[0,1] : x(0) \in [-3,4], |x(t)-x(s)| \leq d |t^2-s^2|, \forall t,s \in C[0,1]\}$. Let $y \in C[0,1]$ and $f : K \to \mathbb{R}$ defined as $f(x)= \int_0^1 x(t)y(t)dt$. Show ...
0
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0answers
14 views

Inequality in the proof of Weak Harnack Inequality

Let $\Omega \subset \mathbb{R}^{n}$ a bounded domain s.t $B_{1} \subset \Omega$ , $u \in H^{1}(\Omega)$ a nonnegative supersolution in the weak sense of the equation $Lu=-D_{i}(a_{ij}(x)D_{j}u)$ ...
5
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0answers
34 views

Variation of the Kempner series

It is easy to argue that the Kempner series converges : $$ \sum\limits_{\substack{n \text{ : 9 is}\\\text{ not a digit of } n}} \frac{1}{n} < \infty$$ Let $E \subset \Bbb N_{>0}$ the subset ...
0
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2answers
37 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 ...
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0answers
10 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
61 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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2answers
47 views

homogeneous differential equations $y' = f(y/x)$

There is a weird Theorem that comes about when considering whether a function is homogeneous (in the sense of the title definition). I was unable to prove it, or to find a proof to it. Can any one ...
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3answers
37 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
16 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
24 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
22 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
20 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
38 views

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

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
votes
3answers
97 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
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1answer
14 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
votes
1answer
12 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} ...
0
votes
1answer
43 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
vote
1answer
32 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
vote
1answer
17 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
53 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 ...