$1.$ Prove the following statements from first principles, using the definition of big$-$O $($in is a symbol shown in the image$)$:

$($a$)3$n$^3+10$n$^2+6$ in O$($n$3^)$;

$($b$)$ k$^$n in O$($n$!),$ for any positive integer k;

$($c$)$ if f$($n$)$ in O$($g$($n$))$ and g$($n$)$ in O$($h$($n$))$ then f$($n$)$ in O$($h$($n$)).$

$($d$)2^$n in $/$ O$($n$^$k$)$ for any finite k$.$