Product of Sequence of Fermat Numbers plus 2

Theorem
Let $F_n$ denote the $n$th Fermat number.

Then:

Proof
The proof proceeds by induction.

For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
 * $F_n = \displaystyle \prod_{j \mathop = 0}^{n - 1} F_j + 2$

Basis for the Induction
$\map P 1$ is the case:

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

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $F_k = \displaystyle \prod_{j \mathop = 0}^{k - 1} F_j + 2$

from which it is to be shown that:
 * $F_{k + 1} = \displaystyle \prod_{j \mathop = 0}^k F_j + 2$

Induction Step
This is the induction step:

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

Therefore:
 * $\forall n \in \Z_{>0}: F_n = \displaystyle \prod_{j \mathop = 0}^{n - 1} F_j + 2$