# Tagged Questions

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### a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
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### Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that lim(Arctan nx) = (pi/2)sgn x for x in R, x>=0. I have a final coming up and I've started doing some of the ...
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### A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
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### let E=C[X] be a normed space and T∈ L(E)… prove that.. [on hold]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
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### Let $\,E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. … [on hold]

Let $E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. Let us consider the identity $I :X→Y$. Prove that I is continuous and bijective. Calculate $\,||I||$. Prove that $I^{-1}$ is ...
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### How we can show that Dirichlet function is measurable with respect to borelian algebra? [closed]

Satisfying the definition of measurability? But, how we can show it?
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### How to find the $n$ zeros of $\displaystyle1+z^n$?

How to find the $n$ zeros of $1+z^n$?
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### show that if $\displaystyle\lim_{n \to \infty} f(n+x)=0$ then $\displaystyle\lim_{x \to \infty}f(x)=0$ [duplicate]

Let $f : \left[0,\infty\right]\to \mathbb R$ be uniformly continuous. If $\displaystyle\lim_{n \to \infty} f(n+x)=0$ where $x$ is in $[0,1]$ then $\displaystyle\lim_{x \to \infty}f(x)=0$ ...
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### A question about $C^2$ domain.

Let $\Omega$ be a $C^2$ domain and assume that $0 \in \partial \Omega$ and that $e_n$ is orthogonal to the boundary of $\Omega$ at $0$. Then in a neighbourhood of $0$, we can put ...
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### Limit and integral properties of a continuous function

Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$. Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$. I've tried ...
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### $\prod_{i = 1}^n x_i^{y_i} \leq y \cdot x$

I need to prove $\prod_{i = 1}^n x_i^{y_i} \leq y \cdot x$ where $x_i \geq 0$ for all $i$ and fixed $y$ where $\sum y_i = 1$. I have looked around and all the proofs I've found have used concavity of ...
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### Confusion about compact subsets of metric spaces being closed

In Rudin's Analysis, we have Theorem: Compact subsets of metric spaces are closed. Can't I generate a counterexample? $\mathbb R$ is a metric space. $(0,1)\in\mathbb R$ is a subset which is ...
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Any hints for this question , My attempt; Say $f(x):0$$\rightarrow$$\mathbb{R}$ The by MVT, there exists a $c$$\in$$(0,\infty)$ , such that; $f'(c)=$$\frac{f(x)-f(0)}{x-0} but im not sure about this ... 2answers 283 views ### Real roots of a polynomial Let p be an even degree polynomial with real coefficients such that the product of the constant term and the leading coefficient is negative. Show that p has at least two real roots. Thanks! 1answer 42 views ### Differentiability of a convex function Let f,g\colon \mathbb{R}\rightarrow \mathbb{R} be convex functions such that f\ge g and f(0)=g(0). Show that if f is differentiable in 0, then g is too and$$ f'(0)=g'(0)$$I have no idea ... 0answers 20 views ### A sequence made from a function under a certain rule. I have been working on this question . My attemt at this question so far:$(a)$From the diagrams I figured out the pattern to be as follows$x_{0},f(x_{0}),f(f(x_{0})),....$wherin$x_{0}=x_{0}$... 3answers 73 views ### Closed subset of compact set is compact If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ... 2answers 72 views ### a question about improper integral, I cannot solve it! If$f(x)$is continuous in$[0,\infty)$, and$\displaystyle\int_c^{\infty}\frac{f(x)}{x}\, dx$is convergent for any$c>0$. Please prove$\displaystyle\int_0^{\infty}\frac{f(\alpha x)-f(\beta ...
If $f(x) > 0$ is continuous at $[0, +\infty]$ and $\displaystyle \int_0^{+\infty} \frac{1}{f(x)} dx$ is convergent, please prove \$\displaystyle \lim_{\lambda \to \infty} \frac{1}{\lambda} ...