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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41 views

Prove an altered p-norm is increasing

$x=(x_1, x_2, \ldots, x_n)$ Prove that $g(p)=[(1/n)(\sum_{k=1}^n |x_k|^p)]^{1/p}$ is increasing on the interval $(0, \infty)$, and find $\lim_{p\to\infty}g(p)$ I find this is extremely difficult. I ...
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12 views

Riemann Integration - Upper Sum

Let $P = \{x_o, x_1, ..., x_n\}$ and $Q = \{x_o, x_1, ...x_j, z, x_{j+1}, x_n\}$ be partitions of $[A, B]$. Note that Q is a refinement of P with just one extra point. Show that if $f: [A,B] \to ...
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46 views

Fourier transform of $e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} $

I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of ...
4
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2answers
121 views

Real solutions of $x^n + y^n = (x+y)^n$

I have to find all real solutions of the following equation: $x^n + y^n = (x+y)^n$ Clearly for $n = 1$, the equation holds for every $x,y$ real numbers. If $n$ is greater or equal to $2$, we do ...
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5answers
86 views

Limit problems and quandaries: finding $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$.

Find $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$. What I did is: $\lim_\limits{n\to \infty }{({n^2-n\over n^2+1})^{n+10}}=\lim_\limits{n\to \infty } {({n^2+1-1-n\over ...
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1answer
40 views

Prove that $\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$

If $f_n(x)=nxe^{-nx^2}~\forall~n=1,2,\cdots$ and $x$ real, show that $$\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$$ Attempt: By the $Mn$ Test, it ...
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2answers
31 views

Does this supremum equal infinity?

This is a generalization of the previous question Does this infinum tend to infinity? Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function satisfying $$\sup ...
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1answer
32 views

Courant. Real numbers determined by nested sequences of rational intervals.

In his book Introduction to Calculus and Analysis vol.1, page 95 Courant writes: Every nested sequence of intervals with real end points contains a real number. To prove this, consider closed ...
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56 views

$ \exists c \in( a, b) \text{ such that } f(c)=\max\limits_{x \in [a, b]} f (x) $

I saw in a corrected. if We have $ f $ continuous on $ [a, b] $ with $ f (a) = f (b) $ and $ f $ differentiable left and right at $ (a,b)$ can we say that $$ \exists c \in (a,b) \text{ such that } ...
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1answer
24 views

Limsup, is there an alternative definition or am I missing the spirit of the question?

Let $X$ be the positive integers Let $H$ be $\mathcal{P}(X)$ For finite $E\in H$ $v(E)$ is the number of points in $E$. Define: ...
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1answer
34 views

nth derivative of ${1\over x}$. A problem. [on hold]

$f(x)=f^{(0)}(x)=x^{-1}$, $f^{(1)}(x)=-x^{-2}$, $f^{(2)}(x)=2x^{-3}$. Therefore, $f^{(n)}(x)=(-1)^{n}n!x^{-n-1}$. Except I see in some places that the expression is different, using, for example, ...
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2answers
40 views

IVT question involving polynomial with even degree

Let $M(x)$ be an even polynomial with a positive leading coefficient, with $a_{2n} > 0, n\ge1 $. Show that there exists a constant $a*\in \mathbb{R}$ such that $M(x)+a = 0$ has a real root if ...
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1answer
18 views

Given the supremum of a set, prove the supremum of a related set

Let S be a bounded set of positive real numbers, and let T = {s$^2$ : s $\in$ S}. Set u:= sup S. Prove that u$^2$ = sup T. Let S,T be as described. Let s$^2$ $\in$ T. Then s $\in$ S and s $\le$ u. ...
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0answers
22 views

Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?

Does anyone know of a simple proof that $\sum_{n=0}^{\infty}a_{n}$ converges whenever the real sequence $\{ a_{n} \}_{n=0}^{\infty}$ satisfies these two conditions? $a_{n}=O(1/n)$; $\lim_{x\uparrow ...
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1answer
17 views

How to prove the period of a trigonometric function decreases?

Let's imagine I have a function $f(x)=\sin x$. We know that the period is going to be $2\pi$. Same idea when we think of $\cos x$. What I am saying is, that the period as $x \to \infty$ remains the ...
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1answer
19 views

What is the oscillation of a function?

Define the oscillation of a function at a point $x$ to be (for an open interval $I$): $$\omega_f(x)=\inf_{x\in I}\sup_{s,t\in I}|f(t)-f(s)|$$ I am a bit confused about the definition above. How am I ...
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1answer
26 views
+50

$U(f+g) \leq U(f)+U(g)$ proof Upper Riemann Integral

$U$ here represents the upper Riemann Integral. I understand the vast majority of this proof, however the part underlined in orange states $\forall \varepsilon>0 $ should it not be ...
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1answer
19 views

Find the extrema of a function

I'm trying to find the extrema of the function : $$ f : \bigg[\frac{1}{2},4\bigg] \rightarrow \mathbb{R}$$ $$f(x)=x^{\ln(\frac{1}{x})}$$ I tried to derivate the function in order to deduce the ...
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0answers
29 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. Any hint would be appreciated! Thanks ...
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4answers
61 views

2003 Putnam A-1 Help needed about sequences

Okay so for $n=1$ there is only one way. For $n=2$ you have, $1+1, 2 + 0$ for $n=3$ you have: $1+1+1, 1+ 2, 3 + 0$ three ways. So $P(n): n$ ways, we must prove the $P(n+1): n + 1$ statement is ...
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1answer
30 views

Using set theory to prove a function problem

I begin with: $$A = \{a \le x < x_0 | f(x) = 0 \}$$ $$B = \{x_0 < x \le b | f(x) = 0 \}$$ Let $c = \sup A$ and let $d = \sup B$ First to prove $f(x) > 0$ for $x \in (c, d)$ I will ...
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19 views

Leading up to Young's Inequality

I am trying to prove Young's Inequality by considering the function $$h(u) = \frac{u^p}{p} + \frac{C^q}{qu^q}$$ for $C,u>0$ and $p,q >1$. We also require $$\frac{1}{p}+\frac{1}{q}=1$$ so that ...
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1answer
50 views

${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$

Let $f\in \mathbb{R}^{I}$ $I$ interval of $\mathbb{R}$ Show that $${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$$ in ...
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8 views

convex region with 7 vertices

Let, $X$ be a convex region in the plane bounded by straight lines. Let, $X$ have $7$ vertices. Suppose $f(x,y)=ax+by+c$ has maximum value $M$ & minimum value $N$ on $X$ & $N<M$. Let, ...
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3answers
40 views

Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$.

Find $f^{(n)}(1)$ where $f(x)={1\over x(2-x)}$. What I did so far: $f(x)=(x(2-x))^{-1}$. $f'(x)=-(x(2-x))^{-2}[2-2x]$ $f''(x)=2(x(2-x))^{-3}[2-2x]^2+2(x(2-x))^{-2}$. It confuses me a lot. I know I ...
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1answer
26 views

Does there exist a non-negative valued compactly supported function such that its Fourier transform only vanishes at a given point?

My question is as follows: Given $t_0\in\mathbb{R}$. Does there exist a non-negative valued compactly supported function $f\in L^1(\mathbb{R})$ such that its Fourier transform, $\widehat f\left( t ...
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1answer
16 views

which of the following sequences $\{f_n\}\in C[0,1]$ must contain a uniformly convergent subsequence?

Could anyone tell me which of the following sequences $\{f_n\}\in C[0,1]$ must contain a uniformly convergent subsequence? $|f_n(t)|\le 3\forall t\in [0,1],\forall n$ $f_n\in C^1[0,1],|f_n(t)|\le ...
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0answers
21 views

contour integral and limit: What is the condition of the interchange the order?

In real real analysis sense, the interchange between limit and integral is hold when integrand is uniformly converges. $i.e$ \begin{align} \int \lim f = \lim \int f \end{align} Here i want to ...
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1answer
26 views

Algebra vs. Sigma-Algebra Condition

I just wanted to clarify the difference between the Algebra and $\sigma$-algebra: Algebra: If $A_1, A_2 \ldots $ are in $\mathscr A$, then $\bigcup_{i = 1}^{n} A_i \in \mathscr A$ ...
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0answers
17 views

Comparison theorem for parabolic partial differential equations

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $J\subseteq\mathbb{R}$ be an intervall $T\in(0,\infty)$ and $f\in C^0\left(\overline{\Omega}\times[0,T]\times J\right)$ be locally Lipschitz ...
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0answers
15 views

Prove there is an $a>0$ such that $\forall x\in [0,1]$, $f(x)>x+a$.

Let $f$ be continuous on $[0,1]$ and $f(x)>x\space \space \forall x \in [0,1]$. Prove there exists an $a>0$ such that $f(x)>x+a\space \space \forall x \in [0,1]$. It is really important ...
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0answers
14 views

Prove that sup and inf of closed and bounded set is in the set.

Assume that $F$ is a non-empty, closed and bounded subset of $\mathbb{R}$, with $d(x,y) = |x-y|$. Show that $supF \in F$ and $infF \in F$.
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1answer
25 views

Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
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1answer
50 views
+50

Continuity on $[a,b]$ implies uniform continuity on $[a,b]$

I don't understand the step underlined in green. I understand that for any $n$ , $|f(x_n)-f(y_n)|\geq \varepsilon$ where $x_n, y_n$ satisfy the conditions given regarding a function being not ...
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0answers
68 views

Question about the application of continuous functions and IVT

I came across a question which says: Suppose that $f:[0,2 \pi] \to \mathbb{R}$ is continuous, and $f(0)=f(2 \pi)$. 1.Show that there exists $x \in [0,\pi]$ such that $f(x)=f(x+ \pi)$. ...
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1answer
40 views

Prove that if ${\{{a_n}^2}\}$ converges (${\{a_n}\}$ is monotone), thus ${\{a_n}\}$ converges and to what?

From Fitzpatrick's Advanced Calculus book: "Suppose that the sequence ${\{a_n}\}$ is monotone, i.e., either monotonically increasing or decreasing. Prove that ${\{a_n}\}$ converges if and only if ...
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1answer
56 views

Find $\lim_\limits{n\to \infty}\{en!\}$.

Find the limit $\lim_\limits{n\to \infty}\{en!\}$. $Attempt:$ $\lim_\limits{n\to \infty}\{en!\}=\lim_\limits{n\to \infty}\{(1+{1\over 1!}+{1\over 2!}+{1\over 3!}+...+{1\over n!}+...)n!\}$. The ...
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2answers
26 views

Property of a differentiable function

Which one of the following is true: 1.If a function real valued function $f$ satisfies $|f(x)-f(y)|\leq |x-y|^{\sqrt2}$ for all $x,y\in \mathbb R$ is $f$ a constant? 2.If $f$ is ...
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3answers
30 views

Find range of $p$ such that the series converges

let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent.Find range of $p$ such that the series $\sum |a_n|^p$ is convergent. My try: To show the series it is ...
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3answers
41 views

Finding $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$

Find $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$. I tried using l'Hospital rule with the continuity of $e$ function. Also tried using Taylor expansion with no success. What ...
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1answer
28 views

Isolate points of a metric space with some properties?

Suppose that all dense subspace of a metric space $(X,d)$ is open. Prove that the set of the isolate points of $X$ is dense in $X$. My Idea: all isolate points of $X$ are in any dense subspace, ...
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1answer
89 views

Theorem $4.3.12$ on ( Mathématiques en BCPST Tome 1 Pascal BEAUGENDRE )

Let $f\in \mathbb{R}^{I}$ and $x_0 \in \stackrel{\ \circ}{I} = \mathrm{int}(A)$ Show that $$\left.\begin{matrix} f \text{ is continuous at } x_0 \\ f \text{ is differentiable at all } x \in I ...
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26 views

This question is based on sum of series. [on hold]

What will be result of $\sum\left ( 1+\left ( \frac{\lambda }{\xi } \right )^{-1} \right )$ please help to find it.
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2answers
76 views

Proving a subset of $l_2$ is closed

Let $l_2$ be the set of all real sequences $x=(x_n)$ such that $\sum|{x_n}|^2 <\infty$ and define the norm $||x_n||_2=(\sum\limits_{n=1}^{\infty}|x_n|)^{\frac{1}{2}}$. I want to show that $A=\{ ...
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3answers
31 views

$\delta-\epsilon$ Question on Ordered Field $\mathbb{R}$

I got came across this question with the $\delta-\epsilon$ definition of a limit, but I do not know how to use it to solve the context of this problem: Problem: Let $f:\mathbb{R}\to\mathbb{R}$ be ...
4
votes
1answer
52 views

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing?

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing? In process of solving this problem, I faced to the problem of proving that $A::$: ...
0
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1answer
39 views

How do I prove this statement about the operator norm?

I stumbled across this equation in a paper, which may seem obvious, but I'm wondering if someone can explain why this is true? By definition of an operator norm, $$\left[(D^*D)^{-1} - ...
0
votes
1answer
43 views

Showing a function is Strictly Decreasing and find the right hand limit

$g(r)=e^{(1/r)(ln( \sum_{i = 1}^{n} |x_i|^r))}$, where all $x_k$ are non-zero. So I have to show that it's strictly decreasing on (0, $\infty$) $g'(r)= (e^{(1/r)(ln( \sum_{i = 1}^{n} ...
0
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2answers
56 views

Showing that stereographic projection is a homeomorphism

For any $n\geq 0$,the unit $n$-sphere is the space $S^{n}\subset \mathbb{R^{n+1}}$ defined by $$S^{n}=S^{n}(1) :=\left\{ (x_{1},...,x_{n+1}) \left\vert\,\sum_{i=1}^{n+1} x_{i}^{2}=1\right.\right\}$$ ...
8
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2answers
712 views

Is the product of two monotone sequences monotone?

Question: The product of monotone sequences is monotone, T or F? Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing. CASE I: Suppose ...