(uppps: this has crossed with pedja's answer)
You begin with an assumtion. You assume, with some unknowns a and b
$\qquad \displaystyle{a \over u} + {b \over u+1 } = {1 \over u(u+1) } $
Then it must be that in the numerator of the product
$\quad \displaystyle a(u+1)+bu = 1 \to (a+b)u+a = 1 \text{ for all } u $
But for all u this can only be possible if $\small (a+b)=0$ and $\small a=1 \to b=-1$, thus you find the only solution for your assumtion
$\qquad \displaystyle{1 \over u} + {-1 \over u+1 } = {1 \over u(u+1) } $