algorithms – Proving existence of an optimal substructure for the DP problem “Sherlock and Cost” from HackerRank

Problem statement from HackerRank ( source: https://www.hackerrank.com/challenges/sherlock-and-cost/problem ) :

In this challenge, you will be given an array $ B $ and must determine an array $ A $ . There is a special rule: For all $ i$, $ A(i) leq B(i) $ , . That is, $ A(i) $ can be any number you choose such that $ 1 leq A(i) leq B(i) $ . Your task is to select a series of $ A(i) $ given $ B(i) $ such that the sum of the absolute difference of consecutive pairs of $ A $ is maximized. This will be the array’s cost, and will be represented by the variable $ S $ below.
The equation can be written: $ S = sum_{i=2}^{N}{ ( A(i) – A(i-1) )} $
For example, if the array $ B = (1,2,3) $ , we know that $ 1 leq A(1) leq 1 $ , $ 1 leq A(2) leq 2 $ , and $ 1 leq A(3) leq 3 $ . Arrays meeting those guidelines are:
(1,1,1), (1,1,2), (1,1,3)
(1,2,1), (1,2,2), (1,2,3)
Our calculations for the arrays are as follows:
|1-1| + |1-1| = 0 , |1-1| + |2-1| = 1, |1-1| + |3-1| = 2
|2-1| + |1-2| = 2 ,|2-1| + |2-2| = 1 ,|2-1| + |3-2| = 2
The maximum value obtained is 2.

My attempt for proving the existence of an optimal substructure:
Denote $ SEQ $ as the set of all sequences $ A $ s.t. $ forall 1 leq i leq n $, $ 1 leq A(i) leq B(i) $. For all $ A in SEQ $ define $ S_A = sum_{i=2}^{n}{ ( A(i) – A(i-1) )} $ .

Let $ hat{A} in SEQ $ s.t. for every $ hat{A’} in SEQ $, $ S_hat{A’} leq S_hat{A} $.
Thus, there exist a sequence of optimal choices $ langle o_1,o_2,…,o_n rangle $ s.t. for all $ 1 leq i leq n $ we chose $ 1 leq alpha leq B(i) $ s.t. $ alpha in B $ and we define $ hat{A}(i) = alpha $, and this is the $ o_i $ choice.
Looking at the sequence of choices $ langle o_1,o_2,…,o_{n-1} rangle $ and looking at the sequence $ tilde A $ that derives from these choices, $ tilde A $ must be optimal. Otherwise, there exist a sequence of choices $ < o_1,…,o_l > $ s.t. $ l < n-1 $ and note that the sequence of choices $ < o_1,…,o_l, o_n > $ give $ hat{A} $. Notice that we have $ l+1 $ choices in the sequence $ < o_1,…,o_l > $ and notice that $ langle o_1,o_2,…,o_n rangle $ is an optimal sequence of choices, but $ l+1 < n $ and since every sequence of choices that yields $ hat{A} $ must have at-least $ n$ choices, this means $ n leq l+1 < n $ , a contradiction.

What do you think about this attempt? how would you prove the existence of an optimal substructure?

Thanks in advance for any help!