there are error in this inductive proof, please point out Proposition. All EECS $203$ students are from the same hometown.

Proof. Let P$($n$)$ be the predicate $$any set of n EECS $203$ students are from the same hometown.$$ We will prove $$n $1$ P$($n$)$ by induction:

Base Case. For n $=1,$ the proposition states that any one EECS $203$ student has the same hometown as themself, which is true.

Inductive Step. Let n $2$ be any integer and assume P$($n$1),$ which states that any set of n$1$ EECS $203$ students all have the same hometown. To prove P$($n$),$ let S $=\{$s$1,...,$sn$\}$ be any set of n EECS $203$ students. Consider the subsets

$$ Sa $=\{$s$1,$s$2...,$sn$1\}$ and$$ Sb $=\{$s$2,$s$3,...,$sn$\}.$

Since Sa has n $1$ students, by the inductive hypothesis, all students in Sa have the same hometown as each other. Since Sb has n $1$ students, also by the inductive hypothesis, all students in Sb have the same hometown as each other. Since $($for example$)$ s$2$ is in both Sa and Sb$,$ that means all students in Sa have the same hometown as s$2,$ and all students in Sb have the same hometown as s$2.$ Therefore all students in Sa $$ Sb $=$ S have the same hometown as s$2,$ and so they have the same hometown as each other, proving P$($n$).$