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, integration through the fundamental theorem of calculus.

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$f_n\rightarrow f$, $g_n\rightarrow g$ in measure,then $f_n.g_n\rightarrow f.g$ in measure

$f_n$,$g_n$ <$\infty$ also, $f_n\rightarrow f$, $g_n\rightarrow g$ in measure,both. how to prove: $f_n.g_n\rightarrow f.g$ in measure a)$\mu X<\infty$ b)$\mu X =\infty$ for case a) : ...
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1answer
8 views

Differentiation of $u(t)=\int_0^t h(s,t)ds, \ \forall t \in \mathbb{R}$ with the multivariable chain rule

Problem: Let $h: \mathbb{R}^2 \to \mathbb{R}$ be continuous and differentiable with respect to its second variable, define $u(t)= \displaystyle \int_0^t h(s,t)ds, \ t \in \mathbb{R}$ In an ...
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2answers
41 views

Assume $f : [0, 1] \rightarrow [0, 1]$ is continuous. Show that there must be a point $x \in [0, 1]$ such that $f(x) = x$

Assume $f : [0, 1] \rightarrow [0, 1]$ is continuous. Show that there must be a point $x \in [0, 1]$ such that $f(x) = x$ I am not even sure how to begin with this problem, I know that $f$ is ...
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1answer
23 views

Is $\sum \cos(n \pi) \frac{n}{n^2+1}$ conditionally or absolutely convergent?

Is $\sum \cos(n \pi) \frac{n}{n^2+1}$ conditionally or absolutely convergent? My Working Clearly, $\sum \cos(n \pi) \frac{n}{n^2+1} = \sum (-1)^n \frac{n}{n^2+1}$, and it can be shown by using ...
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0answers
12 views

prove that F is dense in C(X×Y,R) ? any help!

Let $X,Y$ be compact metric spaces. Let $$F= \left\{ \sum A_i f_i(x) g_i(y), \; f_i \in C(X,\mathbb{R}), \; g_i \in C(Y,\mathbb{R}), \; 1 \leq i \leq n \right\}.$$ Prove that $F$ is dense in $C(X ...
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0answers
20 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
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0answers
21 views

An integral of a sequence of functions

$\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$. Does $$ \liminf_{k\rightarrow\infty} \int_{\Omega} \rho(u_k)\,dx \geq \int_{\Omega} \liminf_{k\rightarrow\infty} ...
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0answers
8 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
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$f_n\rightarrow g$ in $L_1$ and $f_n\rightarrow h$ in $L_2$ .Then $g=h $almost everywhere

$f_n\rightarrow g$ converges in $L_1$ and $f_n\rightarrow h$ converges in $L_2$ how to show: $g=h$ almost everywhere Attempt: convergent in $L_1$ implies convergent in $L_2$. then by triangle ...
4
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1answer
22 views

Divergent subsequence of an unbounded sequence

Let $(a_n)$ be a sequence of real numbers that is unbounded above. Show that $\exists$ a subsequence $(a_{n_k})_{k \ge 1}$ such that $\lim_{k \rightarrow \infty} a_{n_k} = + \infty$. Working ...
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1answer
10 views

Question about Riemann Integration and the indicator function

Let $S \subseteq \mathbb{R}^n$. Suppose $\chi_S$ is integrable and $\int_Q \chi_S = 1 $ for some rectangle $Q$ such that $S \subseteq Q $. Let $\epsilon > 0 $ be given, I want to ask how can I ...
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0answers
15 views

Weierstrass Caratheodory on open interval

Hi all, I have been working on this question for a while now, and if I have understood it correctly shouldn't the answer be that $\phi_{c}=f'(x)$ for all $x \in (a,b)$ as the function f , is now ...
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1answer
50 views

Theorem with an example

I have this theorem In the paper they give an example: But here $H_1$ is not satisfied ! How to correct it please? http://mathoverflow.net/questions/163788/theorem-with-an-example
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1answer
26 views

Taylor's Theorem and inequalities on some interval of the domain?

From the following form of Taylor's Theorem and assuming that $|f(x)|\le 1$ and $|f''(x)|\le 1$ hold on $[0,2]$, $$f(a+h) = f(a) + hf'(a) + (1/2)h^2f''(a+θh),$$ some application of Taylor's Theorem ...
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1answer
19 views

How to interchange sum and integral when measure is in terms of dirac measure?

Let $\{c_{k}\}_{k\in \mathbb Z} \subset \mathbb C$ such that, $\sum_{k\in \mathbb Z} |c_{k}| < \infty.$ Let $\delta_{k}$ is the unit dirac mass at $k $, we note that $\mu = \sum_{k\in \mathbb Z} ...
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2answers
18 views

Proof of Local Boundedness

How to show that $\ f(x)=x^2$ is locally bounded at $x=2$. I am using the definition of local boundedness that states that a function $\ f$ is locally bounded if at $\ x=c$ there is a $\delta>0$ ...
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2answers
20 views

Help with real analysis proof involving supremum

Let $S\subseteq\Re$ be nonempty. Prove that if a number $u$ in $\Re$ has the properties: (i) for every $n\in N$ the number $u-1/n$ is not an upper bound of $S$, and (ii) for every number $n\in N$, the ...
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1answer
38 views

How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
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1answer
28 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
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2answers
32 views

What is the limit of this function as $(x,y)$ approaches $(0,0)$?

Let the function $f \colon (\mathbf{R}^2 \setminus \{(x,y) \in \mathbf{R}^2 \colon x+y = 0 \}) \to \mathbf{R}$ be defined as follows: $$ f(x,y) \colon= \frac{xy}{x+y}$$ if $(x,y) \in \mathbf{R}^2$ ...
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0answers
15 views

Show a function that is a Lipschitz on the interval $[0, +\infty)$but not uniformly there [on hold]

a function that is Lipschitz on the interval $[0, \infty)$, but not uniformly continuous there.
0
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2answers
34 views

Question about uniform convergence?

Show that if $a > 0$, then the convergence of $(\frac{x}{x+n})$ is uniform on the interval $[0, a]$ but not uniform on the interval $[0,\infty)$. Let $f_n(x) = \frac{x}{x+n}$ and $f(x) = 0$ for ...
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1answer
11 views

Finitely additive function bounded by a measure…

have an elementary measure theory question here I can't seem to get. Suppose $\mu$ is a a measure, and $\nu$ is a finitely additive nonnegative set function such that $\nu(A)\le \mu(A)$ for all $\mu$ ...
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2answers
22 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
3
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0answers
33 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
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1answer
43 views

Suppose $f(x)\in L_1$ - Prove that $\lim_{n\rightarrow\infty}\int_0^\infty f(x)\cos(nx)dx = 0$

Assuming knowledge of the cyclic behavior of $cos(x)$, integration by parts, and $\int_0^{\infty} f<\infty$ is enough here? Consider \begin{align} & \int_0^\infty f(x)\cos(nx)dx = ...
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1answer
25 views

Is this function BV or not?

I have a really stupid question about $BV$ functions, I hope somebody can confirm/destroy my opinion! Consider the set $\Omega:=[-1,1]\times [0,1] = A \cup B \subset \mathbb{R}^2$, where ...
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1answer
37 views

Need help on the limit of this sequence [on hold]

I don't understand why the limit for this is not e^2. Got it thanks!
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64 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
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1answer
37 views

Limit of function defined on the rational numbers

Let $f$ the function defined on $\mathbb Q$ by : $ f(n/m) = n $. I would like to know whether it is true that : $ \forall q\in \mathbb Q - \{ 0 \} \quad \forall R>0 \quad \exists \delta >0\quad ...
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0answers
21 views

packing spheroids/ellipsoids

I am thinking about a problem I'm trying to formulate, from a mathematical analysis viewpoint. Suppose that you have a finite region in 3-space, and you want to populate it with spheroids or even ...
1
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1answer
17 views

Check continuity of linear functionals and find norms

1) $c_{00} \owns (x_n) \mapsto \sum_{n=0}^{\infty} x_n \in \mathbb{K}$ where $c_{00}$ is a space of sequences that are eventually equal to $0$ with sup norm 2) $\ell^\infty \owns (x_n) \mapsto ...
2
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0answers
27 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
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0answers
24 views

Level set of a real valued harmonic fucntion

Let $f$ be a real valued harmonic function defined on a neighborhood $U$ of origin in $\mathbb{R}^2$. And $f$ is such that its gradient vanishes at origin. Then how do i show that the set given by ...
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1answer
24 views

If a subset $E$ of $R^n$ is bounded then E is totally bounded

I am trying to prove the above proposition. The book that I am looking at contains E in a cube of the form T=[−b,b]×⋯×[−b,b] for some large b>0. Then, since any subspace of a totally bounded metric ...
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0answers
17 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
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1answer
16 views

Sequential Criterion for Functional Limits

How can I use the sequential criterion for functional limits to show that the following limit exist and compute the limit: $$\lim_{x \rightarrow 0} \sqrt{|x|}\cos\left(1/x\right) \ \text{for} \ x ...
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3answers
17 views

Negation of continuity applied to a sequence

Show that if it is not true that $\lim_{x \to a} f(x)=l$ then $\exists$ $\epsilon$>0 and a sequence $(x_{n}) \rightarrow a$ as $n \rightarrow \infty$ such that $|f(x_{n})-l| \geq \epsilon$. Now ...
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2answers
32 views

Prove $f(x) = \frac{1}{x^2}$ is uniformly continuous on $[1, \infty]$

I am trying to prove this function is uniformily continuous on $[1, \infty]$, so far i have; $$|f(x) - f(x)| = |\frac{1}{x^2} - \frac{1}{y^2}| = |\frac{(x-y)(x+y)}{x^2y^2}|$$ and then, ...
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21 views

Surface Perpendicular to another Surface?

Show that the surface S of equation is perpendicular to any member of the family of surfaces Sa of equation at the point of intersection (1,-1,2). I think I may have an idea on how to proceed but ...
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0answers
22 views

Convergence of series (putnam training) [on hold]

Does the series $\sum_{n=1}^{\infty} \frac {|\sin n|}{n} $ converge?
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0answers
35 views

A subset E of $R^n$ is totally bounded if and only if E is bounded

I am studying Compactness in metric space with Gamelin and Greene's Introduction to Topology and am confused about lemma 5.4 in the book. A metric space $X$ is totally bounded if for each $e > 0$, ...
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1answer
37 views

Graphs with the property that $f=f^{-1}$

Ive been working on this question and can't seem to progress, I know that for a function to have the property $f=f'$ it must be symmetrical about the line $y=x$,I can't find reasoning behind in the ...
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1answer
38 views

Compute $\int_{0}^x \vert \sin(t)\vert dt$ for $x\in \mathbb{R^+}$

Let $x\in \mathbb{R^+}$, compute $$\int_{0}^x \vert \sin(t)\vert dt$$ I tried like this : $$ \int_{0}^x \vert \sin(t)\vert dt=\int_0^{\lfloor \frac{x}{\pi}\rfloor \pi}\vert \sin(t)\vert ...
2
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1answer
21 views

Partial differentiability with respect to $x$ and $y$ of $\int_0^x z(s,y)ds$ where $\partial_y z \in C^0$

I have the following (not accredited and not mandatory) Exercise: Problem: Let $z : \mathbb{R}^2 \to \mathbb{R}$ be continuous and in respect to its second variable partially differentiable. ...
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0answers
19 views

$C^\omega$ notation for real analytic functions

I've seen the notation $C^\omega$ used for the set of real analytic functions (e.g. on an interval). Where does it come from? What exactly does it mean? What is the reason behind it? Who first used ...
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0answers
25 views

Is Arzela-Ascoli with equicontiuous or uniformly equicontinuous

I am still working on this proof of Arzela-Ascoli but now I noticed that in my statement of the theorem I used ''equicontinuous'' to mean ''uniformly equicontinuous''. At least in the direction I ...
0
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0answers
16 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
2
votes
1answer
80 views

Analysis Problem Help

Let $s_1 = \frac{1}{4}$ and $s_{n+1} = s_n(1-s_n)$ for all $n\geq2$. Show that $\lim_{n\to\infty} ns_n =1.$ Can someone please help me to solve this problem? I have couldn't figure out.
2
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0answers
16 views

Area of intersection of polynomial and exponential functions

I was inspired to explore this by a recent post on the math subreddit, which to my knowledge went nowhere. Consider the families of functions $x^y$ and $y^x$. Given some $y \in \Bbb R$, the roots ...