Big$-$O Justification

Indicate whether the following assertions are true or false, and give a justification:

$($n$*$log$($n$)+2$n$)/3$n$2$ is $\backslash$Omega $(1/$n$)$

$($n$*$log$($n$)+2$n$)/3$n$2$ is O$(1/$n$)$

log$($log$($n$))$ is $\backslash$Omega $(1/$n$10)$

$(2$n$)2+$ n$/4+6$ is O$((2$n$)!)$

$1/($n$50)+$ log$16$ is $\backslash$Omega $(1)$

n$!-$n$1000$ is $\_$O$(2$n $+1)\_$