go through this problem and write a small program $($in the language of C$++,$ without seeing the code available on net $/$ not using any standard library functions to get the solution$),$ using basic programming constructs

Problem Statement: Let A be a finite set of integers. Let S be an integer. Find a non$-$empty sub set B of the set A$,$ if it exists, with the sum of its elements equaling to S$.$

I believe, the problem statement is simple enough to understand.

Once you have a program which works,

$1.$ try to argue why your program is correct, in simple English statements, like a lawyer $/$ attorney in the court.

$2.$ try to think of making it faster. Make notes of your thoughts while doing so and the roadblocks you face.