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prove-lim-sup-A-n-c-lim-inf-A-n-c-




Question Number 205054 by SANOGO last updated on 06/Mar/24
prove  (lim sup(A_n ))^c = lim inf(A_n ^c )
$${prove} \\ $$$$\left({lim}\:{sup}\left({A}_{{n}} \right)\right)^{{c}} =\:{lim}\:{inf}\left({A}_{{n}} ^{{c}} \right) \\ $$
Commented by aleks041103 last updated on 11/Mar/24
BUT  A_n =(1−(−1)^n )((n+1)/4)  A_n : 1,0,2,0,3,0,4,...  A_n = { ((0, 2∣n)),((k, n=2k−1≡1(mod 2))) :}    obv.   limsup A_n →+∞   if c=1(for example):  (limsup A_n )^c →+∞  liminf A_n ^c  = liminf A_n  = 0  ⇒(limsup A_n )^c  ≠ liminf A_n ^c  for c=1  ⇒ the statement is obviously false.    But if A_n  is convergent(or diverges to ±∞) then  limsup A_n = liminf A_n  = lim A_n = A  Since A_n  is convergent ⇒ A_n ^c  also converges  ⇒liminf A_n ^c =lim A_n ^c =A^c     ⇒liminf A_n ^c  = A^c  = (limsup A_n )^c
$${BUT} \\ $$$${A}_{{n}} =\left(\mathrm{1}−\left(−\mathrm{1}\right)^{{n}} \right)\frac{{n}+\mathrm{1}}{\mathrm{4}} \\ $$$${A}_{{n}} :\:\mathrm{1},\mathrm{0},\mathrm{2},\mathrm{0},\mathrm{3},\mathrm{0},\mathrm{4},… \\ $$$${A}_{{n}} =\begin{cases}{\mathrm{0},\:\mathrm{2}\mid{n}}\\{{k},\:{n}=\mathrm{2}{k}−\mathrm{1}\equiv\mathrm{1}\left({mod}\:\mathrm{2}\right)}\end{cases} \\ $$$$ \\ $$$${obv}.\: \\ $$$${limsup}\:{A}_{{n}} \rightarrow+\infty\: \\ $$$${if}\:{c}=\mathrm{1}\left({for}\:{example}\right): \\ $$$$\left({limsup}\:{A}_{{n}} \right)^{{c}} \rightarrow+\infty \\ $$$${liminf}\:{A}_{{n}} ^{{c}} \:=\:{liminf}\:{A}_{{n}} \:=\:\mathrm{0} \\ $$$$\Rightarrow\left({limsup}\:{A}_{{n}} \right)^{{c}} \:\neq\:{liminf}\:{A}_{{n}} ^{{c}} \:{for}\:{c}=\mathrm{1} \\ $$$$\Rightarrow\:{the}\:{statement}\:{is}\:{obviously}\:{false}. \\ $$$$ \\ $$$${But}\:{if}\:{A}_{{n}} \:{is}\:{convergent}\left({or}\:{diverges}\:{to}\:\pm\infty\right)\:{then} \\ $$$${limsup}\:{A}_{{n}} =\:{liminf}\:{A}_{{n}} \:=\:{lim}\:{A}_{{n}} =\:{A} \\ $$$${Since}\:{A}_{{n}} \:{is}\:{convergent}\:\Rightarrow\:{A}_{{n}} ^{{c}} \:{also}\:{converges} \\ $$$$\Rightarrow{liminf}\:{A}_{{n}} ^{{c}} ={lim}\:{A}_{{n}} ^{{c}} ={A}^{{c}} \\ $$$$ \\ $$$$\Rightarrow{liminf}\:{A}_{{n}} ^{{c}} \:=\:{A}^{{c}} \:=\:\left({limsup}\:{A}_{{n}} \right)^{{c}} \\ $$

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