Rearange an expression I am learning induction. At one step I have to show that:
$$ 1 - \frac{1}{(1+n)} + \frac{1}{(n+1)(n+2)} $$
can be transformed to
$$ 1 - \frac{1}{n+2} $$
Theese are the steps for the transformation, but I cant understand them:
$$ 1 - \frac{1}{n+1} + \frac{1}{(n+1)(n+2)} = 1 - \frac{(n+2) -1}{(n+1)(n+2)} = 1 - \frac{n+1}{(n+1)(n+2)} = 1 - \frac{1}{n+2} $$
Could you explain to me what happens there? 

$$\begin{align}1-\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}&=1-\frac{n+2}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)}\\&=1+\frac{-(n+2)}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)}\\&=1+\frac{-(n+2)+1}{(n+1)(n+2)}\\&=1+\frac{-(n+1)}{(n+1)(n+2)}\\&=1+\frac{-1}{n+2}\\&=1-\frac{1}{n+2}\end{align}$$
