Problem $4(20$ points$))$

$(1)$ Suppose that $x[n]=2[n-1]+3[n-2],$ and $h[n]=[n-1]+[n]+[n+1].$ Compute the convolution between $x[n]$ and $h[n].$

$(2)$ Prove that for a linear time$-$invariant discrete$-$time system, its output $y[n]$ is the convolution between its input $x[n]$ and its output $h[n].($Hint: this question is for general $x[n]$ and $h[n]).$