Solution $3($b$)$

Loop Invariant:

$($Hint: Your loop invariant should ensure that the output in the final iteration, i$.$e$.$ decision stage, is the optimal set of items with maximum total value that fits in the Knapsack.$)$

In the reminder of this proof, the loop invariant holds true in every iteration $($decision stage$)$:

Initialization:

Maintenance: Assume the loop invariant holds true at the end of $(k-1{)}^{th}$ iteration. Then, during the $k^{th}$ iteration, let the residual capacity of the Knapsack be $c=C-{\displaystyle \underset{\text{i}\text{i}\text{n}\text{K}\text{n}\text{a}\text{p}\text{s}\text{a}\text{c}\text{k}\text{.}}{\overset{?}{}}}{w}_i.$ next item picked by GreedyFracknapsack$($items$)$

Termination: