Given two problems as follows:

Problem A: Given n non$-$negative numbers a$1,$ a$2,...,$ an and another number W $.$ Is there a

subset S $\{1,2,...,$ n$\}$ such that $$

i in S ai $=$ W $?$

Problem B: Given n items $\{$w$1,$ v$1\},\{$w$2,$ v$2\},...,\{$wn$,$ vn$\},$ where for any item i in $\{1,2,...,$ n$\},$

wi is the non$-$negative weight and vi is the non$-$negative value. Given the capacity P and the

value Q$.$ Is there a subset S $\{1,2,...,$ n$\}$ such that $$

i in S wi $<=$ P and $$

i in S vi $>=$ Q$?$

Now, suppose we already know Problem A is NP$-$hard. Please provide the proof to show

Problem B is also NP$-$hard.