State precisely what is wrong with the following proof:

claim: For all positive integers \(n\), it is the case that, in any set \(S\) consisting of n marbles, all marbles in \(S\) are the same color.

proof: By induction on \(n\). 

Base case (\(n=1\)): in any set \(S\) consisting of a single marble, it is trivially true that all marbles in \(S\) are the same color, because only one marble is in \(S\).

Induction step (\(n>1\)): Assume inductively that the claim is true for all sets of \(n-1\) marbles. Under this assumption, we will prove the claim for all sets of \(n\) marbles. 

Let \(S\) be any set of \(n\) marbles. Denote the marbles in \(S\) as \(M(1), M(2), \ldots, M(n)\). That is, \(M(1)\) is the first marble, and so on. Form two size-\((n-1)\) subsets of \(S\) as follows:

\[S_1 = \{M(1), M(2), ..., M(n-1)\}\]
\[S_2 = \{M(2), M(3), ..., M(n)\}.\]


Since \(S_1\) has \(n-1\) marbles, by the inductive assumption all marbles in \(S_1\) are the same color. 
Thus, \(M(1), M(2), ..., M(n-1)\) are the same color.

Likewise, all marbles in \(S_2\) are the same color. Thus, \(M(n-1)\) and \(M(n)\) are the same color.

Thus, \(M(1), M(2), ..., M(n)\) are all the same color.