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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2answers
8 views

convergent subsequences etc.

It is well known that: A sequence of real numbers converges $\{a_n\}$ to p, if and only if every subsequence $\{a_{n_k}\}$ converges to p. I am wondering if this similar statement holds.: ...
1
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2answers
40 views

Let $\{a_n\}_{n=1}^\infty$ be an infinite sequence. Does there exist an infinite series whose partial sums is $\{a_n\}_{n=1}^\infty$?

Definition: By an Infinite Sequence of real numbers, we shall mean any real valued function whose domain is the set of all positive integers. Definition: By an Infinite Series of real numbers, we ...
1
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0answers
9 views

Decomposing convex set into slices

Let $\Omega$ be a convex quadrilateral with vertices $a_1, a_2, a_3, a_4 \in R^2$. Can it be presented as the union of lines connecting the opposite edges $a_4-a_1$ and $a_3-a_2$ as follows? ...
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1answer
28 views

Question about convergence proof, why has he chosen the parameter this way

In this proof he says that $n > 2k$, but would it work if $n \ge k$, if not, why? If $p>0$ and $\alpha$ is real, then $\displaystyle\lim_{n\to\infty}\frac{n^\alpha}{(1+p)^n}=0$. Proof: ...
2
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2answers
40 views

Banach Fixed Point Theorem problem contradiction

I have a following problem. Let $X = R$, $d(x,y) = |x-y|$, $T(x) = \sqrt{x^2 + 1}$ but $T$ does not have a fixed point. Does this contradict Banach's Fixed Point Theorem? I know that if $X$ ...
-1
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1answer
31 views

What are some applications of real analysis? [duplicate]

What are some applications of real analysis? Can someone post a simple example of how real analysis can solve such problems?
2
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1answer
43 views

Limit of a function

I am trying to find the limit (If it does exist) $\lim_{n\rightarrow\infty}\left(1-|\mathcal{X}|^{-\alpha n}\right)^{2^{nC}\left(1-|\mathcal{X}|^{-\alpha n}\right)}$, where $0<\alpha<1$, ...
1
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1answer
21 views

Greatest lower bound implied least upper bound

In the baby Rudin there's a theorem that every ordered set with the least-upper bound property also has the greatest-lower-bound property. I wonder if the converse is true? I tried to prove it and ...
2
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2answers
44 views

Prove that a set in a metric space cannot be both open and closed.

If I have a metric space $X$, and $E \subset X, E \ne X, E \ne\emptyset$. I want to prove that E cannot be both open and closed. I have two strategies, but I am not able to finish them: I assume ...
6
votes
3answers
95 views

$L=\lim_{x\to\infty}(f(x)+f'(x))$ exists . Which of the following statements is\are correct?

Let $f$ be a continously differentiable function on $\mathbb R$. Suppose that $$L=\lim_{x\to\infty}(f(x)+f'(x))$$ exists. If $0<L<\infty$, then which of the following statements is\are ...
1
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1answer
17 views

Sign of differentiable function near critical point

Background : We know that Brownian path oscillates infinitely often changing signs in any neighbourhood of $0$. I was trying to understand if this property holds because that Brownian paths are not ...
1
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2answers
54 views

Is it true that a continuous function with compact support is uniformly continuous?

I've been trying to prove the given $f:\mathbb R\rightarrow \mathbb C$ continuous with compact support, $f$ is uniformly continuous. I don't know if it's true or not, but it is highly plausible and ...
1
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1answer
18 views

Are normed spaces isodyne?

In general, do all non-empty open subsets of a normed space necessarily have the same cardinality?
10
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4answers
457 views

A question on Terence Tao's representation of Peano Axioms

While reading Terence Tao's book on Analysis I had some questions regarding the implication of the Peano Axioms. After writing the following four axioms (which I will write without changing their ...
0
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1answer
24 views

Using $e^{ix}$ instead of sine and cosine in contour integration

A while ago I asked: Evaluation of $\int_{0}^{\infty} cos(x)/(x^2+1)$ using complex analysis. Instead of using $\cos(z)$ an answerer said that is valid to use $e^{ix}$ How is this valid? I dont ...
1
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2answers
25 views

Trying to show property of fractional part

We have this sequence $$\left\{\{\sqrt n\}\right\}_{n=1}^\infty\;,\;\;\{\sqrt n\}:=\;\text{the fractional part of}\;\;\sqrt n$$ The exercise is to prove it doesn't have a limit, and we get several ...
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0answers
26 views

Dubiety About An Inequality Proof

In Principles of Mathematical Analysis, the author is attempting to demonstrate that, if $x>0$ and $y<z$, then $xy<xz$, which essentially states that multiplying by a positive number does not ...
4
votes
5answers
92 views

$f(0)=0$ and $\lvert\,f^\prime (x)\rvert\leq K\lvert\,f(x)\rvert,$ imply that $f\equiv 0$.

I'll start with the precise statement of the problem: Suppose that $f:[0,b]\to\mathbb{R}$ is differentiable, $\,f(0)=0$, and that there exists a real number $K\geq 0$ such that ...
0
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1answer
33 views

Combinatorial techniques, methods, and ideas in (“undergraduate”) real analysis

This question is dual to Probabilistic techniques, methods, and ideas in ("undergraduate") real analysis: I would like to collect some examples of combinatorial arguments to undergraduate ...
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0answers
37 views

Check if a function is L2

I want to check if a function $f$ defined on $[0,T]$ is a $L_2$ function. What I know is $f$ is a $L_1$ function. (but $f$ could be not bounded) So I want to use an inequality like $$ ...
5
votes
3answers
71 views

Evaluation of $\int_{0}^{\infty} cos(x)/(x^2+1)$ using complex analysis.

Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Using only complex analysis. $$I = \int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = (\frac{1}{2})\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 ...
16
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3answers
212 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
0
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1answer
25 views

A modified version of Poincare inequality

We know the general version of Poincare inequality: $$ \int_\Omega |u-u_\Omega|^2dx\leq C\int_\Omega|\nabla u|^2dx,\quad \forall u\in W^{1,2}(\Omega), $$ where $u_\Omega$ is the average of $u$ over ...
2
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0answers
23 views

Choosing a contour to integrate over.

What are the guidelines for choosing a contour? For example to integrate a real function with a singularity somewhere. What type of contour from Square, keyhole, circle, etc should be chosen for ...
0
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0answers
38 views

Vanishing fourier coefficients

Suppose $g,h \in L^1(\mathbb{R} / 2 \pi)$ with $g(x)=h(nx)$, $n \in \mathbb{Z}$. How can I show that $$\widehat{g}(l)=0 \ \text{for} \ l \not\equiv 0 \ \text{mod} \ n?$$ $\widehat{g}(l)$ is defined ...
0
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1answer
55 views

How to check if functions are integrable?

Consider two functions $$ \int_0^1 \frac{1}{e^x-1} dx $$ and $$ \int_0^1 \frac{1}{(e^x-1)^2} dx $$ How to check if these functions are integrable?
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1answer
36 views

Equivalent condition for continuity of a function

Let $g: [0,+\infty) \rightarrow \mathbb{R}$ be a continuous function and let $f: [0,+\infty) \rightarrow \mathbb{R}$ be defined by \begin{equation} f(t) = \inf \{ s \geq 0 \,|\, g(s) > t\}. ...
4
votes
4answers
145 views

Solutions of $z^6 + 1 = 0$

Solve: $$z^6 + 1 = 0$$ That lie in the top region of the plane. We know that: $$(z^2 + 1)(z^4 - z^2 + 1) = 0$$ $$z = -i, i$$ We need to solve: $$((z^2)^2 - (z)^2 + 1) = 0$$ $$z = \frac{1 \pm ...
5
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2answers
39 views

Sufficient conditions to have $f' = O(f(x)/x)$.

Suppose $f$ a nonnegative real-valued function, non-decreasing, $O(x^m)$ for some $m \in \mathbb{Z}_{\geqslant 0}$ and $C^1$, with $f'$ being monotonic and nonnegative. Are this sufficient conditions ...
4
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3answers
187 views

Integration without complex analysis on rational-improper integral

Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm dx$$ Without the use of complex-analysis. With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis? ...
1
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1answer
32 views

Prove/disprove number of zeros inequality

Having a continuous differentiable function $f(x)$, and denote $Z(\cdot)$ number of zeros (assume real line), and $(\cdot)^\prime$ first derivative, I would like to know if following inequality ...
3
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0answers
16 views

Integral Asymptotics for inhomogenous phase

I'm looking for asymptotics for an integral of the form: $$F(n):=\int_{1/2-i\infty}^{1/2+i\infty} e^{\phi(n,z)}dz$$ where $\phi(n,z)=(n-n^3)\log(1-z)+n^2\log(1+z)-n\log(z)$. One can solve for the ...
0
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0answers
20 views

Convergence of distances in metric space [on hold]

If $(X,d)$ is a metric space, $(a_n)$ and $(b_n)$ are Cauchy sequences in $(X,d)$. How do i show that $(a_n):=d(x_n,y_n)$ converges?
0
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1answer
35 views

Prove supremum has infinite integration

Let $f_n : \mathbb{R}\to [0,1]$ be functions such that: $$\sup_{x\in\mathbb{R}} f_n(x)=\frac{1}{n} and \int_{\mathbb{R}}f_n(x)=1 $$ Set $F(x)=\sup_{n\in\mathbb{N}} f_n (x)$. Prove that ...
7
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2answers
73 views

If f is differentiable on (a,b) continuous at a and f has bounded derivative must f be right differentiable at a?

If $f$ is differentiable on $(a,b)$ continuous at a, and $f$ has bounded derivative, must $f$ be right differentiable at $a$? In case answer to previous question is true, is the statement still ...
1
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4answers
134 views

Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?
1
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1answer
39 views

$L^1$ and $L^2$ norm inequaliy

Consider real valued function $f$ defined on $[0, T]$. L1 norm and L2 norm of function $f$ are given by $$ \|f\|_1=\int_0^T |f(t)| \, dt $$ and $$ \|f\|_2=\sqrt{\int_0^T |f(t)|^2 \, dt } $$ Then we ...
0
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3answers
36 views

Metrics (Distances) on $\mathbb{F}$ Theorem Proof

I had a question regarding a Theorem I had come across that described metrics (distances) on ordered field $\mathbb{F}.$ Here it is: Theorem: If $\mathbb{F}$ is an ordered field, then $d(x,y)=|x-y|$ ...
0
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0answers
19 views

Merely conditionally convergent series [on hold]

What is the definition of 'Merely conditionally convergent series'? Is it exactly same as 'conditionally convergent series'? Or different??
0
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0answers
42 views

A basic O.D.E doubt

Consider the non-autonomous O.D.E $\dot{x}(t) = \int h(x(t),y)\mu(t,dy)=F(x(t),t)$ where $\mu(t,dy) = \delta_{y_n}(dy)$ when $t \in [t_n,t_{n+1})$ where $y_n$ and $t_n$ are given sequences s.t. ...
2
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3answers
59 views

Limit of $a_n$, where $a_1=-\frac32$ and $3a_{n+1}=2+a_n^3$.

Let $a_1=-\frac32$, and $3a_{n+1}=2+a_n^3$. I need to show that $\displaystyle \lim_{n\to \infty} a_n = 1$. I can show that the sequence is monotonically increasing and bounded as follows: By the ...
1
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5answers
71 views

If $a,b\in\mathbb R$ with $a<b$, then there is some rational $r$ with $a<r<b$. [duplicate]

How do you prove this question? I was thinking proving contrapositive. But I was stuck..Thanks guys.
1
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1answer
26 views

singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
0
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1answer
21 views

The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
9
votes
3answers
275 views

Calculating a limit of integral

Computing the limit: $$\lim_{n\rightarrow\infty}\left(\frac{1}{3\pi}\int_\pi^{2\pi}\frac{x}{\arctan(nx)} \ dx\right)^n$$ I made the substiution $t=nx$ then, we have: ...
0
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3answers
48 views

Limit of a function to the power of another function

Is there a theorem in real analysis for $\underset{n\rightarrow \infty}\lim f(n)^{g(n)}$, where $f(n)$ and $g(n)$ are arbitrary functions of $n$? Under what conditions on $f(n)$ and $g(n)$ does the ...
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0answers
30 views

Function continuous at end-points [on hold]

If we have a function $f$ that is absolutely continuous on $(-1,1)$ (and also the derivative of $f$ is absolutely continuous on $(-1,1)$) and we have that for the derivative the following limits ...
0
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1answer
20 views

Question about simple functions as described in Folland's Real Analysis

The statement of the theorem is: If $f=0$ almost everywhere, then $\int f =0$. My question is in the proof in Folland (provided below) it seems that we are using some statement like: If $\phi$ is ...
2
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2answers
30 views

Family bounded in $\mathcal{L}^1$ has limit a.e.

Let $(X, \mathcal{F} , \mu )$ be a measure space. Suppose $\lbrace X_n \rbrace$ is a family of functions in $\mathcal{L}^1$, bounded in $\mathcal{L}^1$ i.e. there exist $K \geq 0 $ such that ...
4
votes
1answer
87 views

If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous?

Question: If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous? Background: I thought of the question when proving that "If a function is periodic and continuous, ...