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