Problem $1.[20$ points$]$ Order the following functions by asymptotic dominance. That is$,$ give

an ordering f$1,$ f$2,...,$ such that for all i $>=1,$ we have fi$($n$)=$ O$($fi$+1($n$)).$

$$ f$($n$)=2$n

$.$

$$ f$($n$)=1000$ n$.$

$$ f$($n$)=$ n

log n

$.$

$$ f$($n$)=$ n

$2$

$.$

$$ f$($n$)=22$

n

$.$

$$ f$($n$)=$ log$($n$).$

$$ f$($n$)=$ log$($log$($n$)).$

$$ f$($n$)=$ n $$ log$($n$).$