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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3k views

How to prove a sequence of a function converges uniformly?

For $n \in \mathbb{N}$, define the formula, $$f_n(x)= x/(2n^2x^2+8),\quad x \in [0,1].$$ Prove that the sequence $f_n$ converges uniformly on $[0,1]$, as $n \to \infty$. I know that the definition ...
2
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2answers
65 views

How to compute the limit of the following integral?

Given $b > 0$, let $g(x)$ be a continuous function defined on $[-b, b$]. What is the following limit? $\lim_{N \rightarrow \infty} \frac{1}{\sqrt{N}} \int_{-b}^b e^{-\frac{Nx^{2}}{2}}g(x)\,dx $ ...
0
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1answer
143 views

a multiple choice question regarding the condition of existence of fixed point

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function.Then which of the following statements are necessarily true? $1$. If $f'(x) \le r <1 $ $\forall x \in \mathbb{R}$ , then $f$ has ...
0
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0answers
40 views

integral on $[a,x]$ is zero for all x implies $f=0$ a.e. [duplicate]

Let $f:[a,b] \to \mathbb{\mathbb{R}\cup\{\pm\infty\}}$ be integrable such that $${\int_{[a,x]}fdm=0}$$ for any $x\in [a,b]$. How can I show that $f=0$ almost everywhere? I don't know where to start.
2
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1answer
125 views

Euclidean Metric satisfying the Triangle Inequality - Is there missing details in the proof given here?

The image below comes from the book Geometry and Topology by Miles Reid and Balazs Szendroi. They prove the Triangle Inequality, which is stated below $(2)$. I am happy with the proof of the ...
2
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1answer
33 views

I need to find the value of $a,b \in \mathbb R$ such that the given limit is true

I am given that $\lim_{x \to \infty} \sqrt[3]{8x^3+ax^2}-bx=1$ need to find the value of $a,b \in \mathbb R$ such that the given limit is true. I was able to work the whole thing out, but I have a ...
3
votes
1answer
73 views

Existence of a Continuous Function

If Z is a normal, second countable space and B is an open subset of Z, is it true that there is some continuous $f: Z \rightarrow \mathbb{R}$ where $f(x) > 0$ for all $x \in B$, but $f(x) = 0 $ for ...
1
vote
2answers
127 views

Cardinality of the borel measurable functions?

Using Lebesgue measurable set which is uncountable, one can show that the cardinality of the set of all Lebesgue measurable functions is $2^\mathbb{R}$ I know that Borel $\sigma$-algebra on ...
3
votes
2answers
117 views

Repeating Square Root Closed Form [duplicate]

I've been thinking about repeating square roots: $\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}$. I wrote a program on my calculator to do it $n$ times and I found that, if $x = y^2 - y$ then ...
0
votes
2answers
80 views

Laplacian inequality in Sobolev space

Is the following assertion true? For all $\alpha>0$ there exists some $\theta \in H^2(\Omega)\cap H_0^1(\Omega)$ such that $\|\frac{\Delta \theta}{\theta}\|_\infty \le \alpha.$ Thanks!
2
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0answers
93 views

Inversion formula for Schwartz-space $\mathcal{S}$.

Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and ...
13
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1answer
166 views

Possible generalizations of $\sum_{k=1}^n \cos k$ being bounded

I just read in this month's MAA Math Horizons the problem: Show that $\sum_{k=1}^n \cos k$ is bounded. After a bit of floundering, I realized that the key is to write this as the real part of ...
7
votes
4answers
1k views

If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$

If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$ Since $(a_n)$ is decreasing and bounded below, by Monotonic Convergence ...
3
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1answer
44 views

An integral estimate

Suppose that $f$ is a continuous function on $(0,+\infty)$ and $f\geq 0$. If we have for some positive $\theta$ that $$\frac{1}{t}\int_{0}^{t}f^{\theta}(s)ds\to 0,\quad t\to\infty $$ then I want to ...
1
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2answers
541 views

Continuous functions uniformly convergent to a function, metric spaces, equivalent conditions

Let $X, \ (Y, d)$ be metric spaces, $f_1, f_2, \ldots \ : X \rightarrow Y$ be continuous functions, $f: X \rightarrow Y$ an arbitrary function. Prove that the following condtions are equivalent: 1) ...
0
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2answers
235 views

Limit function of pointwise convergence is always bounded?

If $\{f_n \colon [a,b] \rightarrow \mathbb R\}$ is a sequence of bounded functions converging pointwise to $f \colon [a,b] \rightarrow \mathbb R$, then $f$ is bounded. Is the statement above ...
-1
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2answers
56 views

Need help with the proof that there exists an $x_0$ such that $f(x_0)=x_0$ for the function defined as $|f(x)-f(y)| \leq L|x-y|$ with $0<L<1$

I need to show that there exists a unique $x_0\in \mathbb R$ such that $f(x_0)=x_0$ for the function defined as $|f(x)-f(y)| \leq L|x-y|$ such that with $0<L<1$ and $f: \mathbb R \to \mathbb R$. ...
0
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2answers
115 views

If a continuous function from $\mathbb{R}$ to $[0,\infty)$ does not tend to zero, is its integral greater or equal than some linear function?

Consider a continuous function $f:\mathbb{R}\rightarrow[0,\infty)$ that does not tend to zero as its argument tends to infinity. Formally, there is some $\varepsilon>0$ such that there does not ...
4
votes
2answers
545 views

Intuition behind the difference between derived sets and closed sets?

I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even ...
4
votes
2answers
164 views

Find f ae-differentiable with $f´\in L^1(0,1)$ but not in $BV$…

Here is a natural question which I didn't find in Measure Theory books: Construct a continuous function $f(x)$ in $[0,1]$ with derivative at ae $x\in(0,1)$, and so that $f'(x)\in L^1(0,1)$, but such ...
1
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1answer
56 views

Fourier Series Convergence

Going over some revision. Not really sure what to do for the last bit of aii) I know at $x = 0$, it will converge to $0$ and at $x = \frac{M}{2}$ it will converge to $1$, I'm not seeing how this ...
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3answers
26 views

Mean value theorem proof with tangent

I am trying to show that: $\tan{x}>x$ for $0<x<\pi/2$. How can I show this? I think I can do something with the fcn $\tan{x}-x$ and it derivative, but how can I use this in a proof? ...
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2answers
56 views

Convergence of $\max_{0\le i\le n}|f(i/n)|$

Suppose that $f\colon [0,1]\to\mathbb R$ is a continuous function. How can I prove that $$\max_{0\le i\le n}\biggl|f\Bigl(\frac in\Bigr)\biggr|\to\sup_{0\le x\le1}|f(x)|$$ as $n\to\infty$? Any help ...
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2answers
102 views

non continuous ivt problem

solving for an ivt for a non-continuous function We take $f : [0,1] \to [0,1]$ a non-increasing function, such that $f(x)≥ f(y)$ whenever $x≤ y$; and we want to prove that there exists $c\in [0,1]$ ...
4
votes
1answer
433 views

A function with countable discontinuities is Borel measurable.

Let $f:[a,b] \to \mathbb{R}$ be bounded with countable discontinuities. Show that $f$ is Borel-measurable. One solution uses the fact that A function on a compact interval [a, b] is Riemann ...
0
votes
1answer
36 views

Proving a unique zero with the MVT

Let $f$ be continuous on $[a,b]$, differentiable on $(a,b)$, and let $f(a)=0$ and $f'(x) \neq 0$ for all $x \in (a,b)$. I need to show that $x=a$ is the only zero of $f(x)$ on $[a,b]$. How could I do ...
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2answers
63 views

Inequality proof with cos

I need to prove for any $x_1$ and $x_2$, $|\cos{x_2}-\cos{x_1}| \leq |x_2-x_1|$. How could I do this? Thanks in advance.
2
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1answer
39 views

Is (a sort of) converse to MVT True or False?

Suppose that $f$ is differentiable in a neighborhood of $c$. Then there are numbers $a$ and $b$ with $a<c<b$ so that $\frac{f(b)-f(a)}{b-a}=f'(c)$. Is this true or false? I'm pretty sure its ...
2
votes
0answers
21 views

(Kleiner) transform preserves smoothness class

Consider the transform of nonnegative continuous concave positive homogenuous of first order function $f(x)$, $x \in \mathbb R^n_+$, $f \not\equiv 0$ given by $$ f^\times(y)= \inf \left\{ \left. ...
1
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1answer
72 views

If $\langle\nabla f(x) - \nabla f(y), x-y\rangle \geq 0$ then $H_f \geq 0$

I am trying to prove that for $f:U \subset \mathbb R^n \to \mathbb R$, if $\langle\nabla f(x) - \nabla f(y), x-y\rangle \geq 0$ $\forall x,y \in U$ then $H_f \geq 0$ where $\langle\cdot,\cdot\rangle$ ...
1
vote
1answer
74 views

Convergence of $x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$

$x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$ with $x_0=a\ge0$ I calculated some values for $x_n$ with $a=0,1,2$ and recognized its monotonically decreasing (It converges to $0$). Now the standard approach ...
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1answer
31 views

Finding multivariable solutions given the limit

Find $n,mER$ such that: $limit_{x\rightarrow\infty}$ of $(8x^3+mx^2)^{1/3}-nx$ = 1 Attempt (after multiplying by the conjugate): $\frac{(8x^3+mx^2)-n^2x^2)}{(8x^3+mx^2)^{1/3}+nx)}$ = 1 When I ...
0
votes
1answer
168 views

Continuity and uniformly continuous proof

Prove that the function $f(x)=x^2$ is continuous but not uniformly continuous on the interval $I=(0,\infty)$. I always get confused proving uniform continuity because it is so similar to ...
0
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2answers
69 views

Are these converging or diverging

I am having trouble working out the convergence of these series and was wondering if I could please have some assistance a) $\displaystyle\sum_{n=0}^\infty\sin(e^n)\frac{n}{n^3+1}$ and b) ...
0
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1answer
39 views

How can prove this ineqaulity $||x^{\delta}_{n}-R_{n}y||\le a_{n}|\alpha^{\delta}-\alpha|$

prove that: $$||x^{\delta}_{n}-R_{n}y||\le a_{n}|\alpha^{\delta}-\alpha|$$ where $$x^{\delta}_{n}-R_{n}y=\sum_{j=1}^{n}(\alpha^{\delta}_{j}-\alpha_{j})x_{j}$$ ...
1
vote
1answer
94 views

Compact set and closed set proof

Let $X,Y$ be closed subsets of $\mathbb{R}^p$ and let $X$ be compact. Prove that the set $$X+Y=\{x+y:x\in X, y\in Y\}$$ is closed. I know in order to prove this I must show that $X+Y$ contains ...
1
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1answer
85 views

Subsequence that converges

Suppose that $(x_n)$ is a sequence of real numbers and suppose that the $L_n$ are real numbers such that $L_n\to L$ as $n\to \infty$. If for each $k\ge 1$ there is a subsequence of $(x_n)$ ...
2
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1answer
61 views

$\delta - \epsilon$ argument

Let $f(x)=1/x^2$. Use a $\delta - \epsilon$ argument to prove that for each $x \neq 0$, the quotient $$\frac{f(x+h)-f(x)}{h}$$ converges to $-2x^{-3}$ as $h \to 0$. Can some explain a sketch of the ...
4
votes
2answers
64 views

norm of inverse less than 1

I just wanna ask if what I am doing here make sense: Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
1
vote
1answer
31 views

Definition of Derivative variations

Assuming $f'(c)$, I need to find the following limits: $$\lim_{h\to 0} \frac{f(c-h)-f(c)}{h}$$ $$\lim_{h\to 0} \frac{f(c+nh)-f(c)}{h}, \quad n \neq 0$$ $$\lim_{h\to 0} \frac{f(c+h^2)-f(c)}{h^2}$$ ...
1
vote
2answers
70 views

Function continuity and differentiability

I am given: $$f(x) = \begin{cases} x^{4/3}\sin(1/x) & \text{if $x\neq0$} \\ 0 & \text{if $x=0$} \\ \end{cases} $$ I need to show that $f$ is cts on $\mathbb{R}$ $f$ is differentiable ...
0
votes
1answer
35 views

Solving for a Tangent line

Suppose $f$ and $g$ are differentiable nonzero functions in a neighborhood of $x=2$, and $h=f/g$. I need to determine the line tangent to the graph of $y=g(x)$ at the point $(2,g(2))$ if the tangent ...
2
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2answers
299 views

Analytic functions of a real variable which do not extend to an open complex neighborhood

Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
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2answers
164 views

Is $L^\infty(\mu)$ a locally compact Hausdorff space?

Here $\mu$ is a probability measure. Another similar question is: is it a $\sigma$-compact space? Thank you in advance!
0
votes
1answer
124 views

relation between Holder continuous and weakly differentiable for the coefficients of a pde

I am reading a book on pdes and the author gives the definition of the weak solution using the adjoint operator. For that expression to make sense, for the case of second order elliptic equation one ...
1
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2answers
594 views

Distance between closed and compact sets.

This question is (1-21)(b) from M. Spivak's Calculus on Manifolds. Question: If $A$ is closed, $B$ is compact, and $A \cap B = \emptyset$, prove that there is $d > 0$ such that $||y - x|| \geq d$ ...
2
votes
1answer
160 views

Help with calculation of an integral

How can I evaluate this integral $$\int_0^{+\infty}\frac{e^{-r}}{1+r^2}dr$$ ?
4
votes
2answers
177 views

Does $g$ behave like $t^k$ near the origin?

I asked a similar question early ago, however, my function satisfies more hypothesis, so I am gonna ask it again, please be patient with me. Suppose $G:[0,\infty]\to [0,\infty]$ defined by ...
3
votes
1answer
127 views

Outer Measure Question

Prove or give a counter example: For every open set $U$ of $\mathbb{R}$, $m^*(\bar{U} - U ) = 0.$ My first impression was that it was true, since if $U$ is an open set in $\mathbb{R}$, then it can be ...
4
votes
3answers
61 views

Prove that $f$ has an absolute maximum on $\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. Suppose also that $\displaystyle\lim_{x\to\infty}f(x)=\lim_{x\to-\infty}f(x)=0$ while $f(0)=1$. Prove that $f$ has an absolute maximum on $\mathbb{R}$.