Upper Bound for Lucas Number

Theorem
Let $L_n$ denote the $n$th Lucas number.

Then:
 * $L_n < \paren {\dfrac 7 4}^n$

Proof
The proof proceeds by complete induction.

For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
 * $L_n < \paren {\dfrac 7 4}^n$

$\map P 1$ is the case:

Thus $\map P 1$ is seen to hold.

Basis for the Induction
$\map P 2$ is the case:

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that if $\map P j$ is true, for all $j$ such that $0 \le j \le k$, then it logically follows that $\map P {k + 1}$ is true.

This is the induction hypothesis:
 * $L_k < \paren {\dfrac 7 4}^k$

from which it is to be shown that:
 * $L_{k + 1} < \paren {\dfrac 7 4}^{k + 1}$

Induction Step
This is the induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Second Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_{\ge 1}: L_n < \paren {\dfrac 7 4}^n$