Difficulties in a proof by mathematical induction (2) 
Possible Duplicate:
proof by induction: n/(n+1) 

Continuing from here, I got a splendid answer that helped a lot. I'm tackling one now, but I've run into problems.
Prove by mathematical induction that
$$
\sum_{r=1}^n \frac{1}{r(r+1)} = \frac{n}{n+1}
$$
Since I'm not all that conversant with Tex. . . I got to . . .
$$
\frac{2}{n(n+2)}
$$
How I got there is through fractional addition, expanding then factoring. I'm stuck now. 
Thanks. I would prefer simple answers. 

Hint $\ $ By telescopic induction the proof reduces to the elementary school calculation
$$\rm F(n)\:\! =\:\! \frac{n}{n+1}\ \ \Rightarrow\ \ F(n) - F(n-1) \!\:=\!\: \frac{1}{n(n+1)}$$
