A partial fraction decomposition

$$\prod\limits_{i=0}^{n}{1 \over x + i} = {1 \over n!}\sum\limits_{i=0}^{n}{(-1)^i {n \choose i} \over x + i}$$

For example when $n = 2$,
$${1 \over x(x + 1)(x + 2)} = {1 \over 2}({1 \over x} - {2 \over x + 1} + {1 \over x + 2})$$

I have no proof yet.