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Contraction Mapping Theorem. Prove $\{ y_{1},f(y_{1}),f(f(y_{1})),\ \ldots)...
Let $f$ be a function defined on all of $R$. Assume there is a constant $c$ such that $0< c <1$ and $ |f(x)\ -f(y)\leq c|x-y|$ for all $x,\ y\in R$.(a) Show that $f$ is continuous on $R$ for all...
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Proof feedback for Exercise 2.2.3 Analysis I by Tao for a self taught...
I am self teaching myself from Analysis I book and I would greatly appreciate some feedback on my proofs since I would like to ensure I am tracking along okay. I am new to this so take note of that but...
View ArticleA simple roadmap to explain the proof in baby rudin theorem 2.43 [closed]
Many have been confused about the proof for theorem 2.43 in baby rudin, and I didn't find it satisfying when I searched it in MSE.Here may serve as a simple explanation:Two facts:The initial thought is...
View ArticleBorel measurability of density 0 set [closed]
Suppose $C$ is a Borel set in $\mathbb{R}^n$. Consider the set$$Q=\{x|\lim_{r\to 0}\frac{\mathscr{L}^n(B(x,r)\cap C)}{r^n}=0\}$$How can I show that $Q$ is a Borel set? Thx.
View ArticleShow divergence of a real series. [closed]
Let $a_n = \int_0^1 (1 - x ^ 2) ^ n dx$ for $ n\ge 1$.Show that $ \sum_{n=1} ^\infty a_n $ diverges.Here we can show that {$a_n$} converges to zero by using monotonic convergence theorem as $a_n$ is a...
View Articlecheck whether a set is a positively invariant set
Given a continuous function $V(x(t))$. Suppose $V(x(t)) = 0$ and $\frac{dV}{dt} = 0$ when $x(t) \in A$ with A a compact set, and $V(x(t)) > 0$ and $\frac{dV}{dt} < 0$ when $x(t) \in...
View ArticleShow that a finite limit of $(1-c+f(x))^x$ exists
Consider the function $g(x)=(1-c+f(x))^x$ where $x\in\mathbb{R}_{++}$ and $c\geq0$ but $f(x)$ does not have a closed-form representation. However, I know that $f(x)\geq0$, $f'(x)>0$ and...
View ArticleIs there an intuitive proof of $\sin(x + y)= \sin x\cos y+ \sin y\cos x$ when...
I saw nice geometric proof of $\sin{(x + y)} = \sin{x}\cos{y} + \sin{y}\cos{x}$ using unit circle. But I can't find proof when $x + y > 90^\circ.$Is there intuitive, "simple" or geometric way to...
View ArticleProving a conjecture on a unusual transformation.
This is a follow-up of my previous question which received a lot of very nice and helpful answers. However one question which was actually of equal importance for me was there not answered. Therefore I...
View Article$\lim\limits_n (f_n(x).g_n(x) )= \lim\limits_n f_n(x) .\lim\limits_n g_n(x)$...
Let $(f_n),(g_n)$ are sequences of functions on $\mathbb R$.I was wondering when $ (f_n),(g_n) $ are uniformly convergent then what the conditions that can make the following statement is...
View ArticleHow to prove a gradient is Lipschitz continuous?
I have read in textbooks the following phrase, "assuming the gradient is locally Lipschitz continuous...", but how does one prove that the gradient is locally Lipschitz continuous?If we have a...
View ArticleFundamental theorem of algebra: a proof for undergrads?
The fundamental theorem of algebra is the statement that a complex polynomial of positive degree has at least one root. I do not know complex analysis but I searched for proofs of the statement and...
View ArticleLet $f : [0,1] \to \mathbb{R}$ be a continuous function. Prove that $...
Proof:Define$$F(x) := \int_0^x f(t)\,dt.$$Since $f$ is continuous, the Fundamental Theorem of Calculus (FTC) tells us:$$F'(x) = f(x), \quad \text{for all } x \in [0,1].$$Now define$$G(x) := \int_{1 -...
View ArticleExpression of $\int_{\mathbb R^n}|s|^{-\alpha}e^{-i\langle a, s\rangle}ds =...
Any reference for$$I_{\alpha,a}=\int_{\mathbb R^n}|s|^{-\alpha}e^{-i\langle a, s\rangle}ds= ?,$$where $0<\alpha<n$ and $a\in\mathbb R^n$. Thank you in advance
View ArticleHow to prove that a sequence forms a Schauder basis for $\ell^\infty$?
Let $\alpha, \beta \in \mathbb{R}$ such that $0 < |\alpha / \beta| < 1$. Define the sequence of vectors in $\ell^\infty$ as follows:$x_1 = ( \alpha , \beta , 0 , \ldots) , x_2 = (0,\alpha , \beta...
View ArticleEvaluating $\int_{1}^{\infty}\frac{(a(x^{2}-1))^{m}}{1+...
Can you help me an approach to solve the integrand given constants $a, b> 0$, and $m, n\in\mathbb{N}$?$$\int_{1}^{\infty}\frac{\left ( a\left ( x^{2}- 1 \right ) \right )^{m}}{1+ x}\frac{\exp\left (...
View ArticleProving a Function is Constant Given "Vanishing Lower Derivatives Along Basis...
Please let me know in the comments if this question doesn't meet community guidelines so I can reformulate it. Duplicates are welcome.I've been working on this problem for quite some time - it seems to...
View ArticleLet $f: \mathbb{C} \to \mathbb{C}$ be linear. Does $f$ have to be continuous...
If $f: \mathbb{C} \to \mathbb{C}$ is linear, how can we show that $f(x) = a \cdot x$ for some $a \in \mathbb{C}$? Is this even true?
View ArticleThe surjective reflection between two Cantor Seth [closed]
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be any two Cantor sets (constructed in Exercise 3.) Show that there exists a function $F:[0,1] \to [0,1]$ with the following properties:$F$ is continuous and...
View ArticleAn application of the implicit function theorem
I have a question on an application of the implicit function theorem.Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a $C^1$ function such that $\frac{\partial f}{\partial y}(x,y) \neq 0$ for every $(x,y) \in...
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