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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6 views

Compact operator on $L^{2}$

Let $K(t,s)$ be a real-valued function of two real variables, and let $T: L^2(a,b) \to L^2(a,b)$ be defined by $(Tf)(t) = \int_{a}^{b} K(t,s) f(s) ds$ where $$K(t,s) = \sum_{j=1}^{n} \phi_{j}(t) ...
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18 views

Solve this problem?

Here I have 2 lists, A and B. I am trying to find the connection items of list A got with items of list B. I know: b) all the items of A a) the 1st item of B b) number range from 0-255 A - B 0 ...
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0answers
4 views

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ almost uniformly?

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure How to show or give an counterexample: $f_n\rightarrow f$ almost uniformly. we believe it is false. since both convergence implies there ...
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0answers
10 views

Similar statements to the sequential criterion for functional limits

The sequential criterion for functional limits states that given a function $f:A \rightarrow \mathbb{R}$ and a limit point $c$ of $A$, the following two statements are equivalent: ...
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2answers
21 views

If $f$ is continuous on $[0, \infty)$ and uniformly continuous on $[b, \infty)$ for some $b > 0$ then $f$ is unif. continuous on $[0, \infty).

Prove that if $f$ is continuous on $[0, \infty)$ and uniformly continuous on $[b, \infty)$ for some $b > 0$ then $f$ is unif. continuous on $[0, \infty). So far I have: Let $A = [b, \infty)$ then ...
2
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3answers
38 views

Help with radius of convergence of a power series.

I need to determine the radius of convergence of the series $\sum_{n=1}^\infty a_nx^n$, where $a_n=a^n+b^n$ and $a,b$ are real numbers. Not sure how to approach this one.
2
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1answer
25 views

how ro prove f(x,y) is integrable in $[a,b]\times[c,d]$

If there exits a $f(x,y)$ in $\mathbb{R}^2$,and if we fix any $x$ in $[a,b]$, then $f(x,y)$ is increasing as $y$ increases. Also, if we fix any $y$ in $[c,d]$,the $f(x,y)$ is increasing as $x$ ...
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1answer
35 views

True/False question regarding continuity

For the two scenarios below, either give an example if such a request is possible, or argue why such an request is impossible. I think the first is possible and second is impossible. However, I can't ...
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1answer
22 views

Limit of a sequence and a closed set

It's a dumb question, but I need to assure myself: If $V$ is a subset of a metric space $W$, then if we take a sequence in $V$ and it has a limit in $W\setminus V$, does it mean that $V$ is not ...
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1answer
17 views

Continuous function Limit Problem

Show that there exists only one continuous function $f(x)$ such that $y = f(x)$ satisfies the Kepler equation $y -\epsilon \sin y = x$ for $0 < \epsilon < 1$.
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1answer
13 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
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0answers
12 views

let E=C[X] be a normed space and T∈ L(E)… prove that.. [on hold]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
7
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1answer
57 views

How to learn inequalities and become good at proving them?

I am taking a real analysis course next year and I want to start slowly preparing for that class now, so I hope you can help me. The class is quite challenging and the fail rate is relatively high. ...
0
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1answer
12 views

Directional Derivative along A Curve

Find the directional derivative of $f(x,y,z) = x^2yz^3$ along the curve with parametric equations $$\begin{align}x & = e^{-t}, \\ y & = 1 + 2 \sin t, \\ z & = t - \cos t, \end{align}$$ at ...
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1answer
22 views

Problem about Closed set and Compact set [on hold]

Prove If C:Compact set, D:Closed set and $C\cap D=\varnothing$, then $$d(C,D)>\exists \delta >0???$$ I need your help. Thank you for reading
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0answers
11 views

$L^p$-limit and pointwise limit

For $p\ge1$, I proved that if $f_n\stackrel{L^p}{\to} f$ and $f_n\to g$ a.e then $f=g$ a.e. But, how about the case $0<p<1$? Is it also true?
5
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1answer
33 views

Sequence Limit Problem: If $0 \leq x_{m+n} \leq x_n + x_m$ then limit of $x_n/n$ exists

If the sequence $\{x_n\}$ satisfies the property that $0 \leq x_{m+n} \leq x_n + x_m$ for all $n$, $m \in \mathbb{N}$ , show that the limit of the sequence $\left\{\frac{x_n}{n}\right\}_n$ exists. ...
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1answer
19 views

measurable function and composition of function

Show that if $f$ is a measurable function and $g$ is a continuous function on $\Bbb R$ then $g\circ f$ is measurable. please tell me how to prove it !
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2answers
36 views

Prove by induction that $(1+x)^n \geq 1+nx$ [duplicate]

Prove by induction that $\forall x \in \mathbb{R}, x \geq -1, \forall n \in \mathbb{N},n \geq 0$ that $$(1+x)^n \geq 1+nx$$ First of all I have a problem with x being a real number, how can I use ...
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2answers
28 views

$\int_2^\infty \left(\frac1{x\log^2x}\right)^p\mathrm dx$ diverges for p>1

I see this question and the answer by joriki. However I cannot understand joriki's argument that $$\int_2^\infty \left(\frac1{x\log^2x}\right)^p\mathrm dx$$ diverges for p>1. So I try to show that ...
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1answer
29 views

Convergence/Divergence of $\displaystyle \sum \frac{n^2}{2^{nr}}$

Determine whether $\displaystyle \sum \frac{n^2}{2^{nr}}$, $r \in \mathbb{R}$, diverges or converges Working: Consider the ratio test: \begin{align*} \lim \left| \frac{(n+1)^2}{2^{(n+1)r}} ...
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1answer
20 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
61 views

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

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 ...
0
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1answer
27 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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14 views

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

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 ...
3
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0answers
24 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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1answer
14 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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1answer
29 views

$f_n\rightarrow g$ 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 ...
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1answer
24 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
11 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 ...
0
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1answer
25 views

Weierstrass Caratheodory on open interval

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 said to be ...
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1answer
56 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
28 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 ...
0
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1answer
23 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
20 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
23 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
39 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
29 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
33 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.
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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
12 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
34 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 = ...
1
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1answer
26 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 ...
0
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1answer
38 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!
6
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0answers
68 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 ...
1
vote
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 ...