Problem $2.(30$ points total$)$ Let $x=($:$x_1,x_2,$dots,$x_n$:$)$ be a sequence of distinct integers

for each $i,$ and $x_i{x}_j$ il $\{$:$ij).$ We say $Z=($:$z_1,z_2,$dots,$z_k$:$)$ is a monotonic

increasing subsequence $($MIS$)$ of $x$ if $Z$ is a subsequence of $x$ and $i=1,2,$dotsk$-1$ana$[i]x{x}_i.{?}^{1}a[i]a[j]a[i]iin\{1,2,$dots,$n\}aa[i]B_i^{1}x=($:$2,10,13,20,3,9$:$)a[5]=2x_5=3($:$2,3$:$)O(n^2)ax{z}_i$ for $i=$

$1,2,$dotsk$-1$

$a$