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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Distance in metric space p_{1}

I need to evaluate distance of point [6,6] and circle $x^2 + y^2 = 25$ in metric space $p_{1}(x,y) = ∑|x_k-y_k|$ (sum metric). I know that I need to count $inf(p_{1}([6,6],X), X $ are points from ...
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1answer
28 views

How to check if this function is semicontinuous

Could you tell me how to check that this functions are semicontinuous? $(X, \tau)$ - topological space, $ \ X \neq \emptyset$, $ \ f: X \rightarrow \bar{\mathbb{R}}$, $ \ \bar{\mathbb{R}} = [- ...
1
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1answer
26 views

Inequality that looks like the reverse Minkowski one (but it's not)

Let $x_1, x_2, x_2, y_1, y_2, y_3 \in \mathbb{R}^{+}$. Is it true that if $x_1^2+x_2^2<x_3^2$ and $y_1^2+y_2^2<y_3^2$ then $(x_1+y_1)^2+(x_2+y_2)^2<(x_3+y_3)^2$ ? Thanks in advance.
2
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1answer
60 views

Determining $\bigcap_{n=1}^{\infty}[-n,3^{-n}]$

I have the problem to determine what $$\bigcap_{n=1}^{\infty}[-n,3^{-n}]$$ is. I thought the solution was $[-1,0)$ however it turned out to be $[-1,0]$. Why is this so when $3^{-n}$ will never reach ...
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3answers
73 views

Proving a metric with absolute value [duplicate]

I need to prove that function $\mathbb R × \mathbb R → \mathbb R $ : $f(x,y) = \frac{|x-y|}{1 + |x-y|}$ is a metric on $\mathbb R$. First two axioms are trivial; it's the triangle inequality which is ...
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46 views

Showing $f$ is monotonely decreasing

Let $\lambda_1 < \lambda_2 < \lambda_3$ and $a_1, a_2, a_3 > 0$. $$f(x) = {a_1 \over {x - \lambda_1}} + {a_2 \over {x - \lambda_2}} + {a_3 \over {x - \lambda_3}}$$ Why does this function ...
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2answers
62 views

Is a ray actually a half-line?

I just started studying elementary geometry with Kiselev's plane geometry book. In §5 of the introduction, the author talks about rays, calling them 'half-lines'. That got me wondering whether an ...
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3answers
327 views

The limit of the derivative of an increasing and bounded function is always $0$?

Let $\,f : \mathbb{R} \rightarrow \mathbb{R}$ be a infinitely differentiable function that is increasing and bounded. Then is it true that $\lim_{x\to \infty}f'(x)=0$?
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1answer
60 views

Investigating the convergence of a series using the comparison limit test

Actually not sure how to approach this... but I may be missing something: Replacing the sequence: $x_{n}=1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}-2\sqrt{n},\,\,\,\, n=1,2,....$ By the ...
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1answer
25 views

Show that: $\exists x \in \mathbb{R}. \left|P(x)\right| = e^x$

Show that: $\exists x \in \mathbb{R}. \left|P(x)\right| = e^x$. where $P(x)$ is a polynomial different from the zero-polynomial. Obviously, for every $y \in (0, \infty)$ there's $x$ such that $e^x ...
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1answer
83 views

Limit of an Integral of Bernstein Polynomials

Let $f(x)$ be a function defined on $[0,1]$. The Bernstein Polynomial is defined as $ B_n (x)=\sum_{i=0}^n \binom{n}{i}f(i/n)x^i (1-x)^{n-i} $. Prove that $\lim_{n\rightarrow \infty} \int_0^1 B_n ...
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2answers
28 views

Strange textbook result

According to my textbook, if lim inf $t_n \neq \infty$, then the set $\{n | t_n < \alpha_0\}$ is infinite for $\alpha_0 >$ lim inf $t_n$ Isn't this incorrect? Doesn't the sequence ...
2
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2answers
42 views

Finding roots of function

Consider the function $$f(x)=(2x-9) \cdot 2 \cdot e^{\frac{x^3}{3}-9x+ \frac{46}{3}}$$ Now, the only root to this function is $x=9/2$ I find it quite easy to find this exact root, I will start by ...
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48 views

A Fundamental Property of Metric Spaces …

Let $(X,d)$ be a metric space and $A\subset X$ and also suppose that $G$ is open in $X$ prove the identity: $$ \overline {G\cap A}=\overline {G\cap \overline A} $$ Proposition: The intersection of ...
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4answers
136 views

Some gamma function questions…

I have shown that $\Gamma(a+1)=a\Gamma(a)$ for all $a>0$. But I'd also like to show the following 2 things: 1) Using the previous fact, I'd like to show that $\lim_{a \to 0^{+}}a\Gamma(a) = ...
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1answer
66 views

Question with Isolated Points

I have a question about isolated points. Here is my definition. A point $a \in A \subseteq \mathbb{R}$ is said to be an isolated point of the set $A$ provided there is an open interval $(c,d)$ such ...
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1answer
54 views

proving a tricky inequality

Could someone explain the steps, from the first line of the proof to the second, i.e. how did $nx_0^{n-1}$ become $x_0^{n-1}$?
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4answers
93 views

Continuity and Differentiation on a interval

$$f(x) = \begin{cases} x\sin(1/x), & \text{if $x$ $\ne$ $0$} \\ 0, & \text{if $x$ = $0$} \\ \end{cases}$$ Is $f$ continuous on $(-1/\pi$, 1/$\pi$)? Is $f$ differentiable on $(-1/\pi$, ...
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29 views

Question about a Topological Property

My definition of a topological property of the real numbers system is: "A property of a subset of the real number's that does not change upon the application of a homeomorphism to that set." I ...
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1answer
58 views

“Duality” for weak $L^p$ spaces

Let $1<p<\infty$. Denote by $L^{p,\infty}$ the weak $L^p$ space in $\mathbb{R}^n$ and let $f\in L^{p,\infty}$ where we define the weak $L^p$ quasinorm as $$\|f\|_{p,\infty} = \sup_{\lambda ...
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1answer
270 views

How to make this polynomial the zero polynomial?(recursively)?

Given a fixed $\beta \in \mathbb{R}$, I want to find the $c_0,...,c_n$ for arbitrary $n \in \mathbb{N}$ such that the polynomial \begin{align}P_n(z):=z(1-z) ...
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1answer
64 views

Unique Minimum of an Integral

Let $f(x)$ be a continuously increasing function on $[a,b]$. Show that $\min_{N \in \mathbb{R} } \int_a^b \left | f(x)-N \right |\,dx=\int_{(a+b)/2}^b f(x)\,dx-\int_a^{(a+b)/2}f(x)\,dx$ if and only if ...
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1answer
53 views

Limes $\lim_{n\rightarrow\infty}n^{3/2}(\arctan(\sqrt{n+1})-\arctan\sqrt{n})$

How to deal with the the following limit: $$\lim_{n\rightarrow\infty}n^{3/2}(\arctan(\sqrt{n+1})-\arctan\sqrt{n})$$ I'm clueless.
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1answer
23 views

Question about dilations of a set and open balls

Let $B(0, \frac{r}{2^n} )$. Is it true that $$ B(0, \frac{r}{2^n} ) = \frac{1}{2^n} B(0, r ) \quad?$$
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39 views

Explaining how a decimal expansion must be recurring

Let $p,q$ be whole numbers with $1 < p < q < 9$. Explain why the decimal expansion of the real number $x= \frac{p}{q}$ must be recurring.
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0answers
84 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
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36 views

How I can construct this bijection?

Assume that there is a bijection between $ℤ/2mℤ$ and the set $\{1,2,...,2m\}$ and there is a bijection between $ℤ/2ℤ$ and the set $\{1,2\}$, so it is possible to construct a bijection from the set ...
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1answer
98 views

Root of order k

Im trying to show that a number $x_0$ a root of order $k$ of the polynomial $p(x)$ if and only if $p(x_0)=p^{'}(x_0)= ... =p^{(k-1)}(x_0)=0$ and $p^k(x_0)\neq0$. Is there an easy way to do this using ...
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1answer
25 views

Relation between a function and its derivatives

If we know that $f$ has derivatives of all orders and for some $a,b \in \mathbb{R}$ we have $$f^{(n+2)}(x) +af^{(n+1)}(x)+bf^{(n)}(x)=0$$ How can we conclude that the taylor expansion of $f$ centered ...
1
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1answer
72 views

Numerical Solution of an Equation with Multiple Roots

Let me consider an equation $f(x)=0$ which I know to have a solution $x=x_0$, but I need to find its another solution. So I might consider finding root for the equation $$\frac{f(x)}{x-x_0}=0$$ but I ...
2
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0answers
43 views

Taylor Polynomial for $1/x$

I'm doing a problem that guides you through proving that the Taylor expansion for $f(x)=\frac{1}{x}$ converges for all $x$ such that $0<x<2$. I have shown that $$f(x) - p_n(x)= ...
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0answers
30 views

Question about subsequential limits and limit superiors

Here's the question: Given a sequence $s_n = \left(1, \frac{1}{2}, 1, \frac{1}{2}, \frac{1}{3}, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \ldots \right)$, find the set of subsequential limits and lim ...
14
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2answers
780 views

Why is analysis called “analysis”?

Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the ...
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89 views

Limit of a certain Lebesgue integral

Can someone help me to show that $\lim_{t \to \infty} \int_{\Bbb R} f(x)\sin(xt)dx =0$ for any Lebesgue integrable function $f$? Side note: How do you make the $t \to \infty$ appear directly under ...
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0answers
51 views

Integration as limit of sum

I have facing trouble regarding the following 2 problems.Please Help. Evaluate: $\displaystyle\lim\limits_{n \to \infty} \left[\sin(\frac{\pi}{n})+\sin(\frac{2\pi}{n})+...+\sin(\frac{n\pi}{n}) ...
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2answers
204 views

What is the completion of this space?

This question asks us to show that $\Bbb R$ with the following metric is not complete: Fix a strictly positive function $f \in L^1(\Bbb R)$, and let $d(x,y)=\left|\int_x^y f(t)dt\right|$. It's easy ...
14
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3answers
568 views

Irrationality of $\pi$ another proof

Proposition. Let $\alpha\in\mathbb{R}$. If there is a sequence of integers $a_n,b_n$ such that $0<|b_n\alpha-a_n|\longrightarrow 0^+$ as $n\longrightarrow \infty$, then $\alpha$ is irrational. ...
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4answers
157 views

A question regarding Frobenius method in ODE

Suppose $b(x),c(x)$ are real functions analytic at $0$. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
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79 views

Limit of sequence of improper integrals

I have no idea. How to solve that $$\lim_{n \to \infty } \int_0^{\infty} \! x^{\frac{n}{n+1}} \cdot e^{-x^2} \, dx. $$
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Exercise on subsequences

Question: For each of the sequences given below, give its set of subsequential limits and lim sup and lim inf $a_n = {(-1)}^n$ $b_n = \frac{1}{n}$ $c_n = n^2$ $d_n = \frac{6n+4}{7n - 3}$ Here are ...
2
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3answers
162 views

differential equation $y''(x)-ay^3(x)+by(x)=0$

Hi I am trying to find a solution $y(x)$ to this non linear differential equation $$ y''(x)-ay^3(x)+by(x)=0. $$ I know a nice solution exists, however how can I go about solving this? I know non ...
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1answer
87 views

Proving functions are uniformly continuous help!

a) Prove that $f(x)=x^{1/4}$ is uniformly continuous on $[0,\infty)$. Show that this method can be extended inductively to any $f(x)=x^{1/p}$ for any $p=2^n$ b) Prove directly from the definition ...
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2answers
23 views

$-1\leq\pm 2x\sqrt{1-x^2}\leq1\;\forall\;x\in[-1,1]$?

How to prove without using functions these two inequalities $\forall\; x\in[-1, 1]$: $$-1\leq 2x\sqrt{1-x^2}\leq1$$ $$-1\leq -2x\sqrt{1-x^2}\leq1$$
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99 views

Calculate $\int_0^1 e^x dx$ as a limit of a sum?

As for now, I've been doing the opposite thing. For a given sum in terms of $n\in\mathbb{N}$ I had to calculate the limit (as $n$ approaches infinity) of that sum by applying: ...
0
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1answer
48 views

Smooth function whose $(n+1)$th derivative is defined only on a proper subset of the domain of the $n$th, and the radius contract to $0$

I'm wondering if there exists such a function, whose $(n+1)$th derivative is defined only on a proper subset of the domain where the nth derivative is defined, and with the property that the diameter ...
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2answers
59 views

a question about integral with parameter variables?

I have a problem proving $$\int_{0}^\infty dx {\left(\int_{0}^\infty e^{-x^2t}\sin t\, dt\right)}=\int_{0}^\infty dt\left( \int_{0}^\infty e^{-x^2t}\sin t\, dx\right)$$. I have been struggling for ...
0
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1answer
136 views

Approximating simple functions by step functions almost uniformly

The title says it all. How can we approximate measurable simple functions by step functions almost uniformly in, say, $[0,1]$? Even with the simplest example, $\chi_{A}$, where $A$ is Lebesgue ...
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1answer
73 views

Sequential Criterion Trouble

Define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = 5x, \; x\in \mathbb{Q} \; \text{ and } x^2 + 6\; x\in n\mathbb{Q}$. Using the sequential Criterion, show that $f$ is discontinuous at $1$, but ...
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1answer
57 views

Irrationality Measure $x\in \mathbb{Q} \Longleftrightarrow \mu(x)=1$

Let $x$ be a real number, and let $R$ be the set of positive real numbers $\mu$ for which $$0<|x-\frac{p}{q}|<\frac{1}{q^{\mu}}$$ has (at most) finitely many solutions $p/q$ for $p$ and $q$ ...
2
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1answer
180 views

Prove there is no strictly increasing function $f$ from rationals to reals

Problem: Prove that there is no strictly increasing function $f: Q \rightarrow R$ such that $f(Q)=R$. You may not use cardinality. Where I am: Since $f$ is strictly increasing, $f$ has an inverse (I ...