$1.$ Prove that algorithm Build$-$Max$-$Heap given on Slide $34$ of Lecture $12$ is correct using the loop invariant method. $[4$ points$]$

$2.$ Illustrate the operation of Build$-$Max$-$Heap on the array A $=[5,3,17,10,84,19,6,22,9].[3$ points$]$

$3.$ We obtained a tighter asymptotic bound on Build$-$Max$-$Heap in class. In doing so$,$ we assumed the following:

$($a$)$ There are at most elements with a height h in a heap with n elements. $[3$ points$][4$ points$]$

Prove $($a$)$ and $($b$)$ above.