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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18 views

Limit points and interior points of the cantor set [on hold]

Please help to prove that every point of the cantor set is a limit point and no point is an interior point ( i.e. it has empty interior ) .
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1answer
38 views

Show that $≺$ is a total ordering

Let $ℕ$ be the set of positive integers. Let $D(n)$ denotes the number of divisors of $n$. We define this binary relation: $n≺m⇔n≤m$ and $D(n)≤D(m)$ where $≤$ is the usual ordering in $ℕ$. Show ...
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0answers
21 views

Limit of a continuous function with a parameter

Let $f(x,\alpha)$ be continuous function on $S=(0,1]\times[0,1]$. Suppose that for every segment $[\alpha,\alpha+\Delta\alpha]\in[0,1]$ there exists $x_0=x_0(\Delta \alpha)$ s.t. for $0<x<x_0$ ...
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2answers
23 views

Comparison of the consequences of uniform convergence between the real and complex variable cases,

In the real variable case, I think that uniform convergence preserves continuity and integrability, i.e., for an integral of a sequence of continuous (or integrable) functions, which converge ...
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0answers
19 views

Commutativity and norms of specific operators (Problem 2.7.10 in Kreyszig's functional analysis book)

This is Problem 2.7.10 from Erwin Kreyszig's Introductory Functional Analysis with Applications. Let $C[0,1]$ denote the normed space of all (real- or complex-valued) functions defined and ...
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1answer
21 views

Is this enough to demonstrate divergence of an improper integral?

The integral in question is $$\int_0^\infty (f(x)-a)^2dx$$ Where f(x) is some continuous function and a is some constant. When we expand the integrand,we end up with an $a^2$ term. We can then ...
4
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1answer
56 views

The equality case of the Schwartz inequality

Question: The fact that $a^2 \geq 0$ $ \forall a \in \mathbb{R}$; elementary as it may seem, is nevertheless the fundamental idea upon which most important inequalities are ultimately based. The ...
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2answers
35 views

Convergence of series of $1/n^x$ - pointwise and uniformly,

Consider the series $$\zeta(x) = \sum_{n\ge 1}\frac {1}{n^x}.$$ For which $x \in[0,\infty)$ does it converge pointwise? On which intervals of $[0,\infty)$ does it converge uniformly? My work: I ...
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7answers
307 views

If $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$

Can't quite finish this proof: Prove that if $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$ $ x^2 +xy +y^2 +xy -xy> 0$ $ (x +y)^2 -xy> 0$ Without loss of generality define $x\geq ...
4
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2answers
42 views

Using Mean Value Theorem to Prove Derivative Greater than Zero

I'm working on a problem where at one point I have to show that for $x\ge a$, $$g (x) = \int_a^x f - (x-a) f \left({a+x \over 2} \right)$$, $g'(x) \ge 0$. Additional information: I know that ...
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1answer
27 views

Showing that a function diverges for large x

Consider the function $$f(x) = \frac{x^{3}}{x^{2}+1}$$ Show that as $x\to\infty$ we have that $f(x)\to\infty$. ie. I want to prove that $$\lim_{x\to\infty} \frac{x^3}{x^2+1} = \infty.$$ So given ...
3
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2answers
47 views

Proving limits using epsilon definition

I want to prove that $$\lim_{x\to\infty}\frac{x}{x^{2}+1} = 0.$$ So I start by saying, given $\varepsilon>0$ I want to find $M>0$ such that $$\forall ...
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1answer
24 views

Hypergeometric Series Convergence

For the hypergeometric series $\sum_1^\infty $ $(a)_n (b)_n \over(c)_n n!$, I am looking for help proving that the series converges for $a+b-c<0$. I can understand divergence for different ...
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2answers
40 views

Proving limits exists using epsilon definition

Proving a limit exists. I have found the limit $\lim_{x\to\infty}\frac{x\sin{x}}{x^{2}+1} = 0$. By the sandwich theorem however I have decided that I want to prove that this is the case, ie. I want ...
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2answers
34 views

Proving a limit with a logarithm exists

Show that the limit of $f(x)=x\log{x}$ tends to $\infty$ as $x\to\infty$. So given a $k>0$ I want to find $M>0$ such that $|f(x)|>k$ whenever $|x|>M$. I am having difficulties in not ...
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2answers
53 views

Proof of Max (x,y)

The problem states that $ \max(x,y) = \dfrac { x+y+|y-x|} {2} $ where $x,y \in \mathbb{R}$ Part 1) Prove that this is true. Part 2) Derive a formula for $\max (x,y,z)$. 1) Intuitively i see this as ...
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2answers
58 views

Is this proof of uncountability of Cantor set true?

To construct Cantor set $C$, start with $I_1=[0,1]$ and define $$E_1=\{0,1\}=\{x:x\text{ is an end point of the set }I_1\}.$$ $\operatorname{card}(E)=\#(E)=2$. After deleting the middle open interval ...
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2answers
52 views

Continuous functions of minimal norm

Let $C$ denote the set of continuos functions on $[0,1]$ with the supremum norm. $M\subset C$ such that $$\displaystyle\int_{0}^{1/2}f(t)\, dt-\int_{1/2}^{1}f(t)\, dt=1,\; \forall f\in M$$ Show ...
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3answers
63 views

Want to ensure my proof is rigourous enough.

Question. Prove: $ 0 \leq x < y $ then $ x^n < y^n$ $ \forall n \in \mathbb{N} $ I'm particularly bad at proving obvious things but here it goes. ( please be super strict on analyzing my proof ...
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1answer
30 views

An equation with multiple solutions: finding the maximum of the function of the solutions.

Possibly, this is a bad (stupid) question, but sometimes some discussion helps. I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed ...
4
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1answer
44 views

Is every continuous function that preserves (ir)rationality a rational function?

Let $f: [a, b]\to \mathbb R$ be a continuous function such that for all $r\in\mathbb R$, $f(r)$ is rational iff $r$ is rational. Does it follow that $f$ is a rational function?
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259 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
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1answer
44 views

Convergence in $L^2$ norm

Given a sequence of measurable functions $f_n: \mathbb{R} \rightarrow \mathbb{R}$ such that $\sup_n \|f_n\|_{L^2} < \infty$ where $f_n \rightarrow f$ almost everywhere, is it true that ...
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1answer
14 views

prove that $F$ is $\mu\times\mathcal{L}$ measurable where $F(n,x)=\frac{(2n+1)^2\sin((2n+1)x)}{(n(n+1))^2}$

Let $\mu$ be the counting measure on $\mathbb{N}$ and $\mathcal{L}$ be the Lebesgue measure on $[0,\pi]$. Define the function $F$ on $\mathbb{N}\times\mathcal{L}$ by ...
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3answers
32 views

Borel measurable functions $f:[0,1]\to[0,\infty)$ which cannot be expressed as pointwise limit of nondecreasing sequence of step functions

An interval in this problem may be open, closed or half open. We regard a singleton $\{a\}$ as being an interval also. A step function is a real valued function on $\mathbb{R}$ which is a linear ...
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1answer
45 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
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1answer
88 views

Given $|f'(x)|\leq r<1$ show that $f(x)=x$ is unique solution

Suppose that $|f'(x)|\leq r<1, \forall x\in R$. How do I show that the equation $f(x)=x$ has a unique solution?
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1answer
32 views

Differential equation: $A(x)y''(x)+A'(x)y'(x)+y(x)/A(x)=0$

So give the differential equation $$A(x)y''(x)+A'(x)y'(x)+\frac{y(x)}{A(x)}=0,$$ with $A(x)$ a known function and $y(x)$ te be determined. What is the solution for this differential equation ? I've ...
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26 views

Prove of limit related to $|f(x)|$

Question: Prove that if $\displaystyle \lim_{x \to a} f(x) = L$ then there is a number $\delta > 0$ and a number $M$ such that $|f(x)|<M$ if $0 < |x - a|< \delta$. This means: For every ...
4
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0answers
27 views

Convergence of $n^{-\gamma}T$ where $T$ a hitting time for uniform rvs, can I use CLT?

Let $X_1,X_2,\dots$ be iid uniform on $\{1,\dots,n\}$ and define $T=\inf\{k:X_k=X_r \text{ for some }r<k\}$. The objective is to figure out when $n^{-\gamma} T$ converges weakly to some ...
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2answers
29 views

To find limit points of $F = \{ \frac{n}{1+x} : 0 \leq x <1 \}$

I need to find limit points of $F$ , Given $F = \{ \frac{n}{1+x} : 0 \leq x <1 \}$, $n \in \Bbb N$. According to me there is no limit point as large values of $n$ are not getting close at any ...
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1answer
22 views

Show whether $N(g)=\|g\|_\infty + \|g'\|_\infty$ and $\|\cdot\|_\infty$ are equivalent on $C[0,1]$

Let $L$ be the linear subspace of $C[0,1]$ is the space of continously differentiable functions. I know I've got to show whether there exists an $a,b>0$ such that: $$a\|x\|_\infty \leq ...
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1answer
59 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
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20 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
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0answers
24 views

The dual of the space $L^\infty$. [duplicate]

As we know the dual of $L^p$s are $L^q$s where $\frac{1}{p} + \frac{1}{q} =1$, and dual of $L^1$ is $L^\infty$. What is dual of the space $L^\infty (E)$ where E is a measurable subset of $\mathbb{R} ^ ...
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2answers
82 views

Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$

Evaluate: $$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$ Using real methods only. I am not sure what to do. I tried finding a power series, which was too ugly. I just need some ...
5
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1answer
77 views

Polynomials with specified ranges in intervals

Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$. I want to find a polynomial $f \in \Bbb R[x]$ such that $$\forall x\in[a,b],\mbox{ ...
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62 views

Does $\int_0^\infty |f'(x)| dx < \infty$ conclude $\lim_{x\to \infty} f(x)<\infty $

$f:[0,\infty) \to \mathbb R $ is $C^1$ and $$\int_0^\infty |f'(x)| dx < \infty$$ then can we prove that $\lim_{x\to \infty} f(x)$ exists and $$\lim_{x\to \infty} f(x)<\infty $$ My attempt: ...
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0answers
32 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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1answer
24 views

Dimension of rectifiable curve

Suppose $\Gamma$ is a rectifiable curve (means a curve with finite length), I want to prove that the Hausdorff measure of the intersection of it with closed subset $A\subset \mathbb{R}$ is 0, i.e ...
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2answers
25 views

Problem 8, Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
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0answers
7 views

Can functions with a non-analytic point always be approximated with power laws around the special point?

I'm interested in continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$ that fail to be analytic at a given point (let's say the origin), while still being analytic in a region surrounding it. ...
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0answers
20 views

Problem 2.7-9 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg: Let $C[0,1]$ denote the set of all (real- or ...
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1answer
58 views

Problems taking the limit in $\int_a^b f=\lim_{c\to a}\int_c^b f$ from definitions

Let $f$ be bounded on $[a,b]$ and Riemann integrable for each $c$ with $a<c<b$. I need to show that $f$ is Riemann integrable on $[a,b]$, and $\int_a^b f=\lim_{c\to a}\int_c^b f$. My ...
2
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2answers
52 views

Proof : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$

I need to proof this : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$ then there is $d \in (a,b)$ so $f'(d)=0$. I'm not sure ...
2
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2answers
38 views

The limit of upper bounds is also an upper bound

Question We have a set E which is a subset of the real numbers. There is a sequence ${x_n}$ such that $\{x_n\} \subseteq E$. Suppose there is another sequence $\{y_n\}$ such that the limit as $n$ ...
6
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2answers
47 views

Show that there is no non-zero polynomial $P(u,v)$ in two variables with real coefficients such that $P(x, \cos x) = 0$ holds for all real $x$

I came across the following real analysis problem while reviewing, and I am genuinely stuck on this one: Show that there is no non-zero polynomial $P(u,v)$ in two variables with real coefficients ...
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3answers
55 views

$f(0)>0$, and $f'(x)\le c \lt 1$ for all $x>0$. Prove that $f(x)= x$ has a solution.

$f:[0, \infty) \to \Bbb R$ is continuous on its domain and differentiable on $(0, \infty)$, $f(0)>0$, and $f'(x)\le c \lt 1$ for all $x>0$. Prove that the equation $f(x)= x$ has a solution in ...
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2answers
40 views

Prove that if $b_n$ converges to $B$ and $B \neq 0$ [on hold]

Prove that if $b_n$ converges to $B$ and $B \neq 0$, then there is a positive real number $M$ and a positive integer $N$ such that if $n \geq N$, then $\left | b_n \right |\geq M$. Any help?
5
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3answers
95 views

Is an open $n$-ball homeomorphic to $\mathbb{R}^{n}$?

I read on Wikipedia the following claim: Any open topological $n$-ball is homeomorphic to the Cartesian space $\mathbb{R}^{n}$. No reason or proof was given. Can someone explain? I did try looking ...