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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18 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 ...
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45 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 ...
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1answer
19 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 ...
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1answer
31 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 ...
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2answers
50 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 ...
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1answer
32 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 ...
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0answers
22 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 ...
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1answer
57 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 $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
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28 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 ...
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2answers
57 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
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2answers
86 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
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1answer
53 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: ...
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1answer
31 views

Why does this follow from the triangle inequality?

Proving that differentiability implies continuity.
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2answers
31 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 ...
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1answer
54 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) ...
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0answers
17 views

Lebesgue measure of region under curve

Let $(X,\Sigma,\mu)$ be a $\sigma$-finte measure space and $f \in L^+(X,\Sigma)$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Theorem: Define the area under the graph of $f$ to be ...
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0answers
60 views

About the gradient of a function in $H^{1}(\Omega)$

let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
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1answer
20 views

How to prove that the steeple function is not uniformly convergent?

In class we encountered this function $$f_n(x)=\begin{cases} n^2x, & 0 \leq x \leq 1/n\\ 2n - n^2x, & 1/n \leq x \leq 2/n\\ 0, & 2/n \leq x \leq1 \end{cases}$$ The prof said ...
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2answers
25 views

About the definition of $L^{\infty}$ norm

Let $\Omega$ a limited domain in $\mathbb{R}^{n}$, the space $L^{\infty}(\Omega)=\{f: \Omega\to\mathbb{R} $ measurable $; ||f||_{L^{\infty}(\Omega)}<\infty\}$. Then if a function $f \in ...
3
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1answer
56 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
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3answers
21 views

Prove the existence of limit of certain sequences.

Problem: Let $0<a_1<b_1$ and $$a_{n+1}=\sqrt{a_n\cdot b_n},b_{n+1}=\frac{a_n+b_n}{2}.$$ Prove that $\{a_n\}$ and $\{b_n\}$ converge to some limit. Attempt: By induction and AM-GM, I can show ...
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1answer
56 views

To refute : a function with one discontinuity point is integrable on $\left[0, 1\right]$

If, let $f: \left[ 0, 1\right] \to \mathbb{R}$ continuous with only one discontinuity point is integrable on $\left[ 0, 1 \right]$. I think this is false but I can't find a example that contradicts ...
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0answers
20 views

Uniformly continuous of a function [duplicate]

Let $f:\mathbb{R}\to \mathbb{R} $ be an uniformly continuous functions, how to prove that there are $a, b $ such that $|f(x)|\leq a|x|+b$ thanks for any suggestions
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3answers
45 views

application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$

application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$ where $1/p + 1/q = 1$ I know this is a straight application of the inequality, but how am I assured that the integral of ...
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0answers
17 views

Verifying a startegy to prove convexity on partial domain

Assume you have the multivariate function $$f(x_1,x_2,..,x_n)$$ where: $x_i>0 \forall i$, and $\sum_i x_i = 1$. I need to show that $f$ is a convex function. My plan is to show that it is ...
2
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1answer
33 views

Show $\cos(x^2)/(1+ x^2)$ is uniformly continuous on $\Bbb R$.

now here's how I did proceed. By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| < δ$ and $x,a$ are elements of $E$ ...
2
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3answers
60 views

Why doesn't the limit $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ exist?

Why is this limit non-existant? $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ I can't seem to find $2$ different paths that would show it is non-existant.
3
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0answers
114 views

How do find the numerical average of $x^x$ from $(-4,-2)$?

I wanted to find the approximate average of all real points in $(x)^{x}$ from $[-4,-2]$. This means I am ignoring all complex points and need average to be a real number. To first solve this I found ...
2
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1answer
38 views

How would you approach the limit $\lim_{z \rightarrow 0} \frac{ \sin ||z||_p}{||z||_p}$? [on hold]

How would I solve the limit $\lim_{z \rightarrow 0} \frac{ \sin ||z||_p}{||z||_p}$ ? Note that $p \in [1,\infty]$.
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0answers
24 views

Proof of Riesz–Markov–Kakutani representation theorem in big Rudin

In big Rudin's book, the proof of Riesz's representation theorem quite confuses me. It states that the definition of $\mu_1(V)=\mbox{sup}\{Lf:f\prec V\}$ and $\mu_2(V)=\mbox{inf}\{\mu_1(U): V\subset ...
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2answers
25 views

Completeness of a certain normed space

let $(X, \| \|$) be the normed linear space of bounded uniformly continuous real valued functions defined on $R$. I need to prove that X is complete (under sup norm). My attempt: Let {$f_n$} be a ...
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1answer
17 views

Showing uniform continuity for two intervals

Show that if $f$ is continuous on $[0, \infty)$ and uniformly continuous $[a, \infty)$ for some positive constant $a$, then $f$ is uniformly continuous on $[0, \infty)$. Here is my attempt at the ...
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0answers
19 views

Prove dimension finiteness for a separable subspace of $L^\infty(0,1)$. [on hold]

Let $X$ be a separable subspace of $L^\infty(0,1)$. How do I prove that $X$ is finite dimensional? Thanks in advance.
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2answers
50 views

Derivative of improper integral.

Having trouble trying to differentiate this. $y(t)=e^{it} + \alpha\int_{t}^{\infty} sin(t-s)\frac{y(s)}{s^2} ds $ ...
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26 views

Example of Lipschitz functions

Give an example of a Lipschitz function $f$ on $[0,\infty)$ such that its square $f^2$ is not Lipschitz. My example is to let $f(x) = x$ for $x \in [0, \infty$) Then $|f(x) - f(u)|$ = $|x - u|$ ...
2
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3answers
50 views

Can you inverse a funcion by rotating it?

In school i sometimes run on some excercises where you need to calculate something that has an inverse function in it but you cannot find the inverse and you need to work your way around it. I know ...
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0answers
23 views

Continuous function of bounded variation that is non-monotone? [on hold]

Construct a continuous function of bounded variation on the interval [0,1] which is not monotone in any subinterval. We can follow the pattern of the Cantor-Lebesgue function loosely. For example, at ...
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3answers
139 views

In the definition of a limit, why do we care about all $\epsilon > 0$?

Definition of $\lim_{x \to a} f(x) = L$: $\forall \epsilon > 0, \exists \delta > 0 s.t. |f(x) - L| < \epsilon$ $ if \ 0 < |x-a| < \delta$ Why can't we weaken the assumption ...
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1answer
11 views

Unique solution of an integral equation in $L^1[0,1]$

Let $h\in L^1[0,1]$. Prove that there is a unique solution (almost everywhere) of the following integral equation: $$f(x)=h(x)+\frac{1}{2}\int_0^x\log(1+f(y)^2)dy$$ The idea is to use the fixed-point ...
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1answer
25 views

How does this result follow?

In my real analysis text, an example of proof by induction is given by proving that for any real $x\ge 0$ and all integers $n\ge 0$ $$(1+x)^n \ge 1+nx+\frac {n(n-1)}2x^2$$ I can follow and understand ...
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0answers
21 views

Integral inequality $L^p$ spaces

I'm trying to solve this problem: Let $1<p<\infty$. Then let $f:(0,\infty)\to [0,\infty]$ a measurable non negative function. It's true the following inequality: $$\int_0^\infty ( ...
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1answer
29 views

Convergence of the sequence $z_1=\frac{3}{2}$ with $z_{n+1}=\sqrt{3z_n-2}$

Prove the convergence of the sequence $(z_n)$ such that : $$ z_1=\frac{3}{2}$$ $$z_{n}=\sqrt{3z_{n-1}-2}$$ for every $ n \geq 2$. Calculate also the limit. I have applied induction: $$\frac{3}{2} ...
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2answers
25 views

On differentiable functions

Consider the integral: $I$= $\int_2^{\infty} f(x)g(x) \mathrm{d}x $, where $f(x)$ and$g(x)$ are nonconstant functions. Assuming that this integral exists, does this necessarily mean that $f(x)$ and ...
0
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1answer
40 views

How to prove $\displaystyle\bigcup^\infty_{k=1}(\bigcap^\infty_{n=1}A_{k,n})\subset\bigcap^\infty_{n=1}(\bigcup^\infty_{k=1}A_{k,n})$

Want to show $$\displaystyle\bigcup^\infty_{k=1}\left(\bigcap^\infty_{n=1}A_{k,n}\right)\subset\bigcap^\infty_{n=1}\left(\bigcup^\infty_{k=1}A_{k,n}\right)$$ Note the bottoms are $k=1,n=1$ and ...
4
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1answer
67 views
+250

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
2
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2answers
35 views

Write the Maclaurin series for function $f\left(x\right)=\frac{1}{3x+1}\:$

We have function $f:\mathbb{R}\rightarrow \mathbb{R}$ with $$f\left(x\right)=\frac{1}{3x+1}\:$$ $$x\in \left(-\frac{1}{3},\infty \right)$$ Write the Maclaurin series for this function. Alright so ...
0
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0answers
39 views

Write the interval [a, b) as the collection of intersections of the form (a,b].

Write the interval [a, b) as the collection of intersections of the form (a,b]. Attempt [a, b) = $\cap_{1}^{\infty}(a-\frac{1}{n},b-\frac{1}{n}]$
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0answers
6 views

Giving two examples of functions with some properties.

This is a question from a list. Obtain two $\mathcal{C}^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying these properties: $f(x)=0 \Leftrightarrow 0\leq x\leq 1$; $g(x)=x$ if $|x|\leq 1$, ...
1
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1answer
22 views

intersection of intervals from 1 to infinity

This is a continuation of my previous quesion concerned with finding $ \cup^\infty_{k=1} S_k $ for $ S_k = (1 − 1/k, 2 + 1/k], k \in N $. And the answer intuitively makes sense, but what about for ...
0
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0answers
24 views

Laplace transform of a definite integral

I'm having some troubles with what follows. I am interested in finding the Laplace transform w.r.t. $x$ of some real-valued, positive, continuous (in general well-behaved) function $f(x,t),x,t>0$. ...