# Tagged Questions

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### is the upper limit projection Borel

Let $M$ space metric compact, $\pi:M\times\mathbb{R}^k\rightarrow M$ projection such that $\pi(x,y)=x$. Let $f_n:M\times\mathbb{R}^k\rightarrow \mathbb{R}$ continuous and ...
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### Proving differentiability using Caratheodory's Lemma

Let $I$ be an open interval and let $c\in I$. Let $f:I\rightarrow\mathbb{R}$ be continuous and define $g:I\rightarrow\mathbb{R}$ by $g(x)=\left|f(x)\right|$. Prove that if $g$ is ...
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### rewriting the inverse image

If $\phi_k:\mathbb{R}^2\rightarrow \mathbb{R}$ are continuous functions, for all $k\geq0$ and $$\phi=\limsup_{n\rightarrow \infty }\phi_n$$ Let $A\subset \mathbb{R}$, is possible to write ...
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### Convex combinations sequence question

Let $a_0,a_1,\beta$ be given with $0<\beta<1$ Let the sequence be defined by $a_{n+2} = \beta a_{n+1} + (1-\beta)a_n$ for $n\geq0$ Show that $\{a_n\}$ converges and find its limit.. How to ...
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### Does existence of global minimum imply coercivity?

It is known that a coercive function over a closed, unbounded set has a global minimum. Is the converse true ? The larger context for this question is the following question: Suppose we are given a ...
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### Linearizing systems about critical points.

$$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$ Find all the critical points of the following systems and derive the linearised system about each ...
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### Find all real numbers $x$ such that $2+\sqrt{|x|}=x$

I'm revising for an exam and my lecture notes and online sheets etc. from my lecturers are very unhelpful, and I've completely forgotten how to do this type of question. So any help would be greatly ...
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### Every Cauchy net is convergent [duplicate]

Prove that in a Banach space every Cauchy net is convergent. I have trouble to prove this, please help.Thanks Edit:Let $A$ be a directed set and $\{f_{\alpha}\}_{\alpha\in A}$ is a net in $X$ ...
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### Lemma. $\exists N\in\mathbb{N}\forall n>N: x_n=0\Leftrightarrow x=\frac{q}{p^N}$ for some $q\in\mathbb{N}.$

https://proofwiki.org/wiki/User:J_D_Bowen/Math710_HW1 in the lemma of exercise 5 in direction $\Leftarrow$: Why $S_{N-1}=x$? I dont see! Thanks!
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### Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
Is there a direct way to see that $p \times id : [0,1]^2 \rightarrow S^1 \times [0,1]$ is a quotient map with $(p \times id)(x,y) = (e^{ix},y)$? By direct way, I mean is there an obvious argument why ...