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.