4. Suppose you have a set $W$ containing $n$ weights, namely 1 gram, 2 grams, 4 grams, 8 grams and so on until $2^{n-1}$ grams.
• Prove that for any natural number between 1 and $2^n - 1$, you can find a subset of $W$ whose total weight in grams is the number chosen. [4 marks]
• Prove that no other set of powers has the same property. [3 marks]
• Explain the mathematical sense of the previous point. [3 marks]