manipulation of subtraction I am trying to solve an induction problem and got stuck at this part.
$$ 1 - \frac{n+2}{(n+2)!} + \frac{n+1}{(n+2)!} = 1 - \frac{(n+2) - (n+1)}{(n+2)!} $$
Shouldn't it be
$$ 1 - \frac{n+2}{(n+2)!} + \frac{n+1}{(n+2)!} = 1 - \frac{(n+2) + (n+1)}{(n+2)!} $$
How do you get the left expression to the right expression?

Note that 
$$-\frac{A+B}{C}=(-1)\times \left(\frac{A+B}{C}\right)=(-1)\times\left(\frac{A}{C}+\frac{B}{C}\right)=-\frac{A}{C}-\frac{B}{C}=\frac{-A-B}{C}.$$
The minus sign in front of the first fraction does influence the plus sign in front of $B$ of the numerator of the first fraction.
