Sum of Sequence of k x k!/Proof 1

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
 * $\displaystyle \sum_{j \mathop = 1}^n j \times j! = \paren {n + 1}! - 1$

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:
 * $\displaystyle \sum_{j \mathop = 1}^k j \times j! = \paren {k + 1}! - 1$

from which it is to be shown that:
 * $\displaystyle \sum_{j \mathop = 1}^{k + 1} j \times j! = \paren {k + 2}! - 1$

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:
 * $\forall n \in \Z_{\ge 1}: \displaystyle \sum_{j \mathop = 1}^n j \times j! = \paren {n + 1}! - 1$