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cauchy schwarz inequality problemes

I have to prove that for all $x,y,z>0$, $$\left(\frac{x+y}{x+y+z}\right)^{0.5} + \left(\frac{x+z}{x+y+z}\right)^{0.5} + \left(\frac{z+y}{x+y+z}\right)^{0.5} \leq 6^{0.5}$$ using Cauchy-Schwarz ...
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Computing $\lim_{x\to 0}[(\sin x)^{\frac{1}{x}}+(\frac{1}{x})^{\sin x}]$

How can we compute the following: $$\lim_{x\to 0}\left[(\sin x)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin x}\right]?$$ The expression looks rather daunting. If $x\to 0$, then $\sin x\to 0$ but ...
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A structure elementarily equivalent to $(\mathbb{N},0,\operatorname{S},<,+,\cdot)$

Given $\mathfrak{R} = (\mathbb{N},0,\operatorname{S},<,+,\cdot)$, Let $$\Sigma = \{ 0 < c, \operatorname{S}{0} < c, \operatorname{S}\operatorname{S}{0} < c, \ldots\}$$ By compactness ...
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Complex function (find u and v)

I don't have an idea how to find the real and imaginary part of the following functions: $$\dfrac{1}{z},~\dfrac{2z-i}{iz+2},~\dfrac{z}{z^2+1},~\ln |z|$$ Can anybody help me?
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Parseval's Theorem Q

I have this question: I know Parseval's theorem is given by $2a_0^2 + \sum_1^{\infty} (a_n^2 + b_n^2) = \frac {2}{T} \int_{-T/2}^{T/2} f(x)^2 dx$, where T is the period. $f(x)$ is even, so I ...
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When compactness implies sequentially compactness [duplicate]

It is well known that if $X$ is a metric space then sequentially compactness and compactness are equivalent.\ Now we consider a normed vector space $E$ and its dual $E^\ast$. From Banach ...
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Find all nonzero matrices $A\in M_2(\Bbb Z_3)$ which are not invertible.

Find all nonzero (with every element $\neq$0) matrices $A\in M_2(\Bbb Z_3)$ which are not invertible, and explain why they aren't invertible. It seems simple indeed, but I'm not sure how to solve ...