Finding Maximum value of consecutive positive integers with constraints
Let, $x,y,z$ be consecutive positive integers such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} > \frac{1}{10}$I need to find the maximum value of $x+y+z$ ?My attempt:I tried guessing first, To...
View ArticleConvergence in distribution implies $P(|X_n| \leq 1) \to P(|X| \leq 1)$?
Let $X, X_1, X_2,X_3,\ldots$ be real-valued random variables on a common probability space. Suppose that $X_n \to X$ in distribution as $X \to \infty$. Is it true that $\lim\limits_{n \to \infty}...
View ArticleFind $d_k$ that minimize $\sum_{k=1}^\infty d_k\cdot\exp(-(d_1+\cdots+d_{k-1}))$
N.B.: The question has been edited after Sangchul Lee's answer to include an extra condition that is required for the originalproblem to make sense.Consider a strictly increasing sequence of strictly...
View Articleproof lower semicontinuous envelope is lower semicontinuous
My professor defined lower semicontinuous envelope$f_{*}(x):=\inf \left\{\liminf _{n \rightarrow+\infty} f\left(x_n\right): x_n \rightarrow x\right\}$.his proof fixed $x_0$ and $x_k\rightarrow x_0$by...
View ArticleCurve starting at vertex of a cone with a given tangent direction stays...
While working on some algebraic stuff, I was led to the following situtation:Assume we have two points $x_0\neq y_0$ in $\mathbb{R}^n$. The vector starting at $x_0$ and ending in $y_0$ is denoted by...
View ArticleProve that the supremum of $\int_{0}^{1}f(x)dx$ is $1$.
I consider the set $E=\{f\in C^0([0,1],\mathbb{R}) : 0\leq f\leq 1, f(0)=0\}$and I want to show that the supremum of $\int_{0}^{1}f(x)\mathrm dx$ over $E$ is $1$.First it is easy to see that for all...
View ArticleGradient of function with error term
Given a function$$f \in C^\infty:R^n\to R \text{ such that } f(x) = \frac{||x||^2}{2} + o(||x||^2) \text{ and }(n \geq 2)$$The lecture I'm taking states that $\nabla f(0) = 0$ and I just don't get...
View ArticleContraction Mapping, why is there a constant? [duplicate]
The same question was asked some time ago here, but it does not clarify a concern I have. So picking up from where they left off:"From Wikipedia, a contraction mapping is a function $f:M\to M$ on a...
View ArticleShow that the constant polynomial $g $ closest to $f$
In the vector space $C(1,3)$, endowed with the internal product $$\langle f, g \rangle = \int_ {1}^{3} f (x) g (x) dx $$ for all $f, g \in C (1,3) $. For $f (x) = \dfrac{1}{x} $ with $x \in (1,3) $,...
View ArticleDoes there exist a $C_c^\infty(\mathbb R)$ function $f$ such that $\sup_x...
I know that for a $C_c^\infty(\mathbb R)$ function $f$, and for each given $k\in\mathbb N$, we can give existence of $M_k$ such that$$\sup_x |x|^k|f^{(k)}(x)|\leq M_k^k$$as $f$ is a Schwartz function....
View ArticleShow that the space of Hilbert-Schmidt operators forms a Banach space
I am having trouble proving the following result:Show that the space $X$ of bounded operators on a separable Hilbertspace into itself for which the Hilbert-Schmidt norm is finite, is aBanach spaceMy...
View ArticleShow that $X_n \to 0$ in probability under given condition.
Let $k > 0$. Suppose that $$\forall \epsilon > 0: \exists N: \forall n \geq N: P(|X_n| \geq \epsilon) \leq \epsilon k$$Show that $X_n \xrightarrow{P}{ 0}$. Attempt:We have to show: $$\forall...
View ArticleAre there any non-trivial subsets of $\mathbb{R}$ that satisfy trichotomy and...
Consider any subset $\mathbb{P}\subset\mathbb{R}.$Trichotomy says that if $x\in\mathbb{R},$ then exactly one of the following holds-$$x\in\mathbb{P}$$$$-x\in\mathbb{P}$$$$x=0$$Additive closure says...
View ArticleFind the spectrum of the compact operator...
Given the operator $T:L^2([0,1])\to L^2([0,1])$ defined by$$T(f)(x)=\int_0^x \frac{f(t)}{1+t^2}dt$$I have to find the spectrum of T, $\sigma(T)$.Since I have already proved that T is compact, by the...
View ArticleWhat are some useful problem solving strategies for real analysis?
In this blog, Professor Tao exhibited some problem solving strategies that can help students in their study of (mostly) measure theory and some are intended for analysis in general. I'd love to see...
View ArticleNon-linear, continuous, strictly increasing, unbounded function with a...
I have a strictly increasing, continuous, non-linear, unbounded (above and below) function $w:\mathbb{R}\rightarrow{} \mathbb{R}$ such that for fixed $x,y\in \mathbb{R}$, it holds that for some...
View Articledivergence of split sequences with $\sum a_i + \sum b_i = \infty \Rightarrow...
Update: I edited the question to: $\sum_{i=1}^{\infty} a_i + \sum_{i=1}^{\infty}\frac{1}{b_i} = \infty $, sorry for inconvinience.A given series $(x_i)_{i=1}^{\infty}$, is split into $(a_i)$ and...
View ArticleWhy do I get two values for the following limit using two different methods...
When I evaluate the integral $$\lim_{x \to 0} \frac{\log(1+x+x^2)+\log(1+x^2-x)}{\sec x-\cos x}$$by using the identity $\lim_{x \to 0} \log(1+x)=x$, it simplifies to $$\lim_{x \to 0}...
View ArticleExistence of solution for the ODE $x' = F(x)$ when $F$ is a $C^\infty$ function
Consider the system of differential equations of form $$x'(t) = F(x(t))$$ where $x(t) \in \mathbb{R}^n$ and the function $F:\mathbb{R}^n \to \mathbb{R}^n$ is a $C^\infty$ function.I am looking for...
View ArticleEstimating the value of cosine with its power series
In Amann's Analysis, the following is stated:Here $\mathbb{R}^{\times}$ denotes $\mathbb{R} \setminus \{ 0 \}$.Corollary $II.7.9$ is the followingHere $s_n$ denotes the $n$th partial sum and $s$ the...
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