Articles of analysis

Proving $C()$ Is Not Complete Under $L_1$ Without A Counter Example

I’d like to show that $C([0,1])$ (that is, the set of functions $\{f:[0,1]\rightarrow \mathbb{R} \, \textrm{ and } \, f \, \textrm{is continuous} \}$ is not a complete mertric space under the $L_1$ distance function: $$ d(f,g) = \int_0^1 |f(x)-g(x)|dx $$ I can find counter examples (for example, here) but would rather prove it using […]

A variation of fundamental lemma of variation of calculus .

I have a question on a variation of the fundamental lemma . If $\int_\Omega f(x) g(x)=0$ and $f, g $ are $C^0\Omega$ functions and $\int_\Omega g(x)=0 $ then is it possible that there exist some constant $c$ such that $f(x)=c$ for all $x\in \Omega$ I tried to use mollification on one of the function and […]

Has the polynomial distinct roots? How can I prove it?

I want to prove that the polynomial $$ f_p(x) = x^{2p+2} – cx^{2p} – dx^p – 1 $$ ,where $c>0$ and $d>0$ are real numbers, has distinct roots. Also $p>0$ is an even integer. How can I prove that the polynomial $f_p(x)$ has distinct roots for any $c$,$d$ and $p$. PS: There is a similar […]

How to prove $r^2=2$ ? (Dedekind's cut)

Let a (Dedekind) cut $r=\{p \in \mathbb{Q} :p^2<2 \text{ or } p<0\}$ and a cut $2^*=\{t\in \mathbb{Q} : t<2\}$. I want to prove $r^2=2^*$. I could show that $r^2 \subset 2^*$ easily, but I couldn’t show that $2^* \subset r^2$. How to show that there is $p$, $p’ \in r$ such that $t \leq pp’ […]

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt{c_n}\leq\lim\sup\sqrt{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here’s Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} \leq \lim_{n\to\infty} \inf \sqrt[n]{c_n},$$ $$ \lim_{n\to\infty} \sup \sqrt[n]{c_n} \leq \lim_{n\to\infty} \sup \frac{c_{n+1}}{c_n}.$$ Now Rudin has given a proof of the second inequality. Here’s my proof of the first. Let $$\alpha […]

$\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}=\prod_{k=1}^{n}\bigl(1-\frac{1}{2k}\bigr)$

i cant see why we have : $$\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}$$ $$\dfrac{(2n-1)!!}{(2n)!!} =\prod_{k=1}^{n}\left(1-\dfrac{1}{2k}\right),$$ Even i see the notion of Double factorial this question is related to that one : Behaviour of the sequence $u_n = \frac{\sqrt{n}}{4^n}\binom{2n}{n}$ For $\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}$ $\dfrac{(2n)!}{4^n n!^2}=\dfrac{(2n)!}{2^{2n} n!^2}=\dfrac{(2n)\times (2n-1)!}{2^{2n} (n\times (n-2)!)^2}$ For $\dfrac{(2n-1)!!}{(2n)!!} =\prod_{k=1}^{n}\left(1-\dfrac{1}{2k}\right),$ note that $n!! = \prod_{i=0}^k […]

Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$

I want to analyze the behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$, i.e I want to show it behaves like $e^x(\frac{2}{x}+\frac{2}{3x^2}+…)$ as $x\rightarrow \infty$ I started by looking the at the exponent $\cos\sqrt{t}$, taking the derivative and setting it $0$ gives me $0,\pi^2,2^2\pi^2,3^2\pi^2,…$ My guess is to use Lapalce Method now but I do not know how. How can […]

Finding $\lim_{x \to +\infty} \left(1+\frac{\cos x}{2\sqrt{x}}\right)$

Let $f(x)=x+\sin(\sqrt{x})$. I want to find $\lim_{x \to +\infty} f'(x)$. Attempt 1 We have $$f'(x)=1+\frac{\cos x}{2\sqrt{x}} \leq 1 + \left|\frac{\cos x}{2\sqrt{x}}\right| \leq 1 + \frac{1}{2\sqrt{x}}.$$ As $x \rightarrow \infty$, $\sqrt{x} \rightarrow \infty$, hence $\frac{1}{2\sqrt{x}} \rightarrow 0$. Then $1 + \frac{1}{2\sqrt{x}} \rightarrow 1$. Therefore by the Sandwich Theorem $f'(x) \rightarrow 1$. Lemma $\lim_{x \to \infty} g(x)=l$ […]

What is the difference between advanced calculus, vector calculus, multivariable calculus, multivariable real analysis and vector analysis?

What is the difference between advanced calculus, vector calculus, multivariable calculus, multivariable real analysis and vector analysis? What I think I know Vector calculus and multivariable calculus are the same. Multivariable real analysis and vector analysis are the same and both are the formalization of multivariable/vector calculus. Am I right? what’s the difference between advanced […]

Inverse function theorem application

I have to solve this question with this solution way. But I made some mistakes while solving. I cannot see thesemistakes. And I cannot reach the wanted result properly. Please somebody helps me. Thank you so much. I Will be happy to help me. —– I added an example from my notebook. I need to […]