Correctness of use induction in the proof

Question

"Let $S$ be a subset of vector space $V$. Let $P_1, ... , P_n$ be elements of vector space $V$. Let $S$ be the set of all linear combinations $t_1 P_1 + ... t_n P_n$, with $0 \le t_i$ and $t_1 + ... t_n = 1$. Show that S is convex."
To prove that I used induction on number of elements. However, I had an argue with my teacher that says that my proof does not prove the statement in the task, but rather something else. After a long discussion, we didn't manage to understand each other. Could you point me, why use of induction is incorrect here?