Equivalence of Definitions of Lucas Numbers

Definition 1 implies Definition 2
Let $\left\langle{L_n}\right\rangle$ be the sequence defined as in definition 1.

It follows from Lucas Number as Sum of Fibonacci Numbers that $\left\langle{L_n}\right\rangle$ is the sequence defined as in definition 2.

Definition 2 implies Definition 1
Proof by induction:

Let $\left\langle{L_n}\right\rangle$ be the sequence defined as in definition 2.

For all $n \in \N$, let $P \left({n}\right)$ be the proposition:
 * $L_n = \begin{cases}

2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$

Basis for the Induction
We have that:
 * $L_0 = F_{-1} + F_1 = 1 + 1 = 2$
 * $L_1 = F_0 + F_2 = 0 + 1 = 1$
 * $L_2 = F_1 + F_3 = 1 + 2 = 3$

Thus $P \left({0}\right)$, $P \left({1}\right)$ and $P \left({2}\right)$ hold.

$P \left({3}\right)$ is the case:


 * $L_3 = F_2 + F_4 = 1 + 3 = 4$

So $P \left({3}\right)$, as $L_3 = L_1 + L_2$.

This is our basis for the induction.

Induction Hypothesis
Let us make the supposition that, for some $k \in \N: k \ge 1$, the proposition $P \left({j}\right)$ holds for all $j \in \N: 1 \le j \le k$.

We shall show that it logically follows that $P \left({k + 1}\right)$ is true.

So this is our induction hypothesis:
 * $\forall 1 \le j \le k: L_j = L_{j - 1} + L_{j - 2}$

Then we need to show:
 * $L_{k + 1} = L_k + L_{k - 1}$

Induction Step
This is our induction step:

Hence $L_n = L_{n - 2} + L_{n - 1}$ follows by the Second Principle of Mathematical Induction.

That is: $\left\langle{L_n}\right\rangle$ is the sequence defined as in definition 1.