Problems with proof by induction $\frac1{1\times2} + \frac1{2\times3} + \dots + \frac1{n(n+1)} = \frac1{n+1}$? 
$$\frac1{1\times2} + \frac1{2\times3} + \dots + \frac1{n(n+1)} = \frac1{n+1}$$

Prove for $n=1$: $$\frac1{1\times2}=\frac1{1+1}=\frac12$$
Hip: $$\frac1{1\times2} + \frac1{2\times3} + \dots + \frac1{n(n+1)} = \frac1{n+1}$$
Demonstration: $$\frac1{n+1} + \frac1{(n+1)(n+2)}=\dots=\frac1{(n+1)+1}$$
My problem is that I can't find the correct algebra steps to get from the beginning of the demonstration to the end of the demonstration. 

$\frac {1}{1\times 2}=1-1/2$
$\frac{1}{2\times 3}=1/2 - 1/3$
........
$\frac {1}{n(n+1)} =1/n- \frac {1}{n+1}$
Add them up and cancel the middle terms to get $1-\frac {1}{n+1}=\frac {n}{n+1}$
