Low Level Well-Ordering Principle Proof Suppose that I have k envelopes, numbered $0,1,...,k−1$, such that envelope i contains $2^i$ dollars. Using the well-ordering principle, prove the following claim.
Claim: for any integer $0 ≤ n < 2^k$, there is a set of envelopes that contain exactly n dollars between them.
How would I tackle this problem? I am stumped. I know that the well-ordering principle states that a non-empty set will always have a smallest element in the set, but I don't exactly know how to apply it to write the proof.

Let S denote the values in $\{0,1,2,…,2k−1\}$ that cannot be expressed with the envelopes. $0$ is not in $S$, because "no envelopes" has zero dollars. $S$ is well-ordered because it is a subset of the naturals; if it is nonempty it has a smallest element $s$. Hence $0,1,2,…,s−1$ can all be expressed with combinations of envelopes, but s cannot be. You need to use the combination of these two facts to get a contradiction; hence $S$ is empty and your theorem is proved.
If you need a hint for how to get that contradiction, let me know.
How do I get to a contradiction?
