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 […]

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 […]

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 […]

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’ […]

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 […]

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 […]

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 […]

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 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 […]

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 […]

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