Principle of Mathematical Induction/Warning/Example 1

Example of Incorrect Use of Principle of Mathematical Induction
Let $L_k$ denote the $k$th Lucas number.

Let $F_k$ denote the $k$th Fibonacci number.

Given that $L_n = F_n$ for $n = 1, 2, \ldots, k$, we see that:

Hence:
 * $\forall n \in \Z_{>0}: F_n = L_n$

Refutation
We have made the assumption that $L_n = F_n$ for $n = 1, 2, \ldots, k$.

However, we have that:
 * $L_2 = 3$

while:
 * $F_2 = 1$

Hence as the base case has been demonstrated to be false, the proof is invalid.