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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1answer
30 views

Ordered Field $\mathbb{F}$ Corollary Proof

I wanted to check my proof for a corollary on ordered fields $\mathbb{F}$. Here is the corollary: Corollary: Let $\mathbb{F}$ be an ordered field and $a\in\mathbb{F}.$ If $a>0$, then ...
2
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1answer
36 views

On a $\epsilon$-$n$ proof of a limit of a sequence of functions.

Put $\delta_n = 2^{-n}$. To each positive integer $n$ and each real number $t$ corresponds a unique integer $k = k_n(t)$ that satisfies $k\delta_n \le t< (k+1) \delta_n$. Define $$ \psi_n(t) = ...
1
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1answer
23 views

Evaluate this limit (dominated convergence theorem),

$\lim_{n\to\infty} x^n$, for $x \in [0,1]$. I'm using the dominated convergence theorem on a few problems and keep running into this issue. What's the limit of the above function? Clearly, for $x$ ...
3
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0answers
50 views

Power Series starting from $n=0$

I have the following product $$\frac{1}{6}\sum_{n=0}^{\infty}\left(\frac{-x}{2}\right)^n\sum_{n=0}^{\infty}\frac{a_nx^{n+2}}{n!}$$ Where $a_n$ is an arbitrary coefficient. If I factor out $x^2$ ...
2
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0answers
31 views

Inequality proof critique

I'm trying to prove an inequality. I think I'm correct but it would be helpful if my proof can be critiqued. Given two matrices of same dimension $T$ and $P$ with $P \geq 0$ and scalar $\delta>0$, ...
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0answers
41 views

Why $x\in C_k$ implies $x/3\in C_{k+1}$.

If $C_k$ is an approximation of the Cantor set, why $x\in C_k$ implies $x/3\in C_{k+1}$. Thanks!
2
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1answer
38 views

Determining a radius convergence of a power series

Let $$ \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1} $$ Is there an immediate way to determine $R=1$?
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0answers
32 views

Integration problem that may use DCT

I am trying to solve the following problem. Let $f \in L^2(0,1)$ and define $$f_n(x)= n \int\limits_{k/n}^{(k+1)/n} f(y) dy $$ for $x \in [k/n, (k+1)/n)$, $k=0,1, \dots, n-1.$ Show that $f_n ...
1
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1answer
26 views

Proof of the starting part of theorem 1.17 Rudin ( Complex and Reals)

The proof I would like is of the following fact: Put $\delta_n = 2^{-n}$. To each positive integer n and each real number t corresponds a unique integer $ k = k_n(t)$ that satisfies $k \delta_n \le t ...
2
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1answer
43 views

Cantor set's endpoints.

Prove that: If $[a,b]$ is one of the closed intervals that makes up one the approximation $C_k$ of the Cantor set then the endpoints $\{a,b\}\subset C$ where $C$ is the cantor set. I should prove ...
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2answers
21 views

If $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure.

I want to prove that if $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure. This is the proof: Suppose not. Then there exist $\epsilon>0,\delta> 0$ such that $μ \{x: ...
4
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1answer
35 views

Absolute continuity of the ratio of two absolutely continuous functions

Will the ratio of two absolutely continuous functions, say $f$ and $g$ where $g$ is non-vanishing, remain absolutely continuous?
1
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0answers
88 views

Prove the Schwarz inequality using $ 2xy \leq x^2 + y^2 $

Im really bad at analysis and this problem was recommend to me to help me grasp some basics of $\epsilon $ $\delta $ So im doing a problem ( though its like 12 pieces ) this is i guess the fourth ...
4
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3answers
248 views

Image of open set is not open?

I'm confused by the proof that $\epsilon$-$\delta$ continuity is equivalent to open-set continuity. One can prove that a function is $\epsilon$-$\delta$-continuous if and only if the preimage of any ...
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2answers
31 views

Bound on and integral

If $\alpha \in \Bbb R$, how can I show $$\int_{-M}^M \frac{1}{\sqrt{|x-\alpha|}} \, dx \le 4 \sqrt{M}$$ For $M>0$, Rewriting the integral gives $$\int_{-M+\alpha}^{M + \alpha} ...
4
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4answers
66 views

$\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ dense in $\mathbb{R}$? [duplicate]

I'm guessing $\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ is dense in $\mathbb{R}$. I'm having a mental block. How do you show that? (This is motivated by a different hypothesis: if $f$ is ...
0
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1answer
28 views

Hint on metric space

I want show that $(a_n):=d(x_n,y_n)$ converges, if $(X,d)$ is a metric space, $(a_n)$ and $(b_n)$ are cauchy sequences in $(X,d)$. Here is what i do; From the hypothesis, $(a_n)$ is bounded, because ...
1
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1answer
44 views

Prove that $\frac{\partial^2 f}{\partial x \partial y}=0$

Supose $g:\mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}$ is a function of class $C^2$, and $\frac{\partial^2g}{\partial x^2}=\frac{\partial^2g}{\partial y^2}$. If we define ...
0
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1answer
42 views

How to define the 'error'

I have true data $G$ and wrong data $F$. Both are $n$ dimension vector. $G\in \{G_i| 0<G_i<255\}, i = 1:n$. Because the ...
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2answers
28 views

Show $\sup_{y>0}\left|\int_0^\infty \int_t^\infty f(x,y) \cos\left(\dfrac{t}{y}\right)dx\,\,dt\right|<\infty$

Suppose $f$ is Lebesgue measurable on $[0,\infty)\times [0,\infty)$ and $g\in L^1([0,\infty))$. If $|xf(x,y)|\leq g(x)$ for all $y\in [0,\infty)$ prove that $$\sup_{y>0}\left|\int_0^\infty ...
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0answers
10 views

a.e convergence of càdlàg function

I'm having troubles with formally prove this: Let $f:\mathbb{R_+} \to \mathbb{R}$ be a piecewise constant and càdlàg function and define $$f^h(x) = ...
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0answers
32 views

How do I see that $|f_n -f|^p = \lim_{n \rightarrow \infty} \inf |f_n - f_{n_k}|$? [on hold]

Let $(X, \mathcal E, \mu)$ be a measure space, $p \in [1, \infty]$ and $f, f_n \in \mathcal M(\mathcal E)$ Suppose $(f_n)$ is Cauchy in $\mu$-p-mean and $f_{n_k} \rightarrow f$ converge ...
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0answers
18 views

Problem with the definition of semi-ring and $\sigma$-sets

I have a problem with a statement I found concerning the definition of semi-ring and that of $\sigma$-set. So, here there is. Assume the definition of a semi-ring $\mathcal{S}$ over a non-empty set ...
1
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1answer
40 views

An unusual two dimensional sum

Can anyone prove or reference a proof for the following bound (unless it's not true!) $$\sum_{|\underline{k}|_{\infty} > M} \frac{1}{((k_1)^2 + (k_2)^2 )^2} \leq \frac{C}{M^2}$$ where ...
7
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1answer
45 views

Differentiating a constant and switching order

Why does this work? $$\int x^2e^{ax}dx = \int \frac{d^2}{da^2}e^{ax}dx = \frac{d^2}{da^2}\int e^{ax}dx = \frac {d^2}{da^2} \frac {e^{ax}}a = \frac{e^{ax}(a^2x^2-2ax+2)}{a^3}$$ $a$ is a constant, so ...
0
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0answers
27 views

What is the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?

Is $1$ the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?
0
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1answer
34 views

Proof of limit of a piecewise function, rational, irrational

Prove that: If $f(x) = 0$ for irrational $x$ and $f(x) = 1$ for rational $x$ then $\lim_{x \to a} f(x)$ does not exist for any $a$. So begin by the opposite assumption: Assume $\lim_{x \to a} f(x) ...
2
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2answers
33 views

Finding the limit of this integral: $\lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$ if $q<p+1$

I am trying to find the following limit provided: $q<p+1$: $$ \lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$$ Dividing by $n x^q$ so we have $$\dfrac{n x^p+x^q}{x^p+n ...
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0answers
11 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
1
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0answers
19 views

$\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$

I was thinking about $\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$ The inspiration came from the following 3 integrals : Lemma If $f(x)$ is a bounded non-negative ...
2
votes
2answers
24 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 ) .
2
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1answer
41 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 ...
1
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0answers
26 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 $0<x<x_0$ ...
1
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2answers
32 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
26 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 ...
0
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1answer
28 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
votes
1answer
65 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 ...
5
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2answers
50 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 ...
5
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7answers
322 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
43 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 ...
1
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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
48 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 ...
1
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1answer
26 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
42 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
37 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 ...
2
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2answers
58 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 ...
1
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2answers
62 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 ...
1
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3answers
64 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 ...
0
votes
1answer
33 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 ...