I want to prove an obviously wrong statement: Any set $\{v_1, \ldots , v_n\}$ of non-zero vectors in Rm is linearly independent.
We proceed by induction on the size of the set. Consider the base case in which we have a single vector $v$. Since $v\neq 0$, the set $\{v\}$ is linearly independent. Now assume that for a fixed $n ∈ N$, any set of $n$ vectors is linearly independent. Consider any set of $n + 1$ vectors, say $\{v_1, \ldots , v_{n+1}\}$. By the inductive hypothesis, the sets $\{v_1, \ldots , v_n\}$ and $\{v_2, \ldots , v_{n+1}\}$ are linearly independent. Therefore, the set $\{v_1, \ldots , v_{n+1}\}$ is also linearly independent. By induction, the claim thus holds for a set of non-zero vectors of any size.