Argue the correctness of $$Build$-$Max$-$Heap$$

Build$-$Max$-$Heap$($A$,$ n$)$

$1$: A$.$heap$-$size $=$ n

$2$: fori$=$n$2$downto$1$do $3$: Max$-$Heapify$($A$,$ i$)$

by proving the following loop invariant: $$At the start of each iteration of the for loop of lines $2-3,$ each node i$+1,$i$+2,...,$n is the root of a max$-$heap$$