Combinatorial proof of a binomial identity

Hi, I'm trying to combinatorially prove the following identity:

$\displaystyle \sum_{i=0}^{r} \binom{n+i}{n}=\binom{n+1+r}{n+1}$

On the right side, $\displaystyle \binom{n+1+r}{n+1}$ is the number of ways we can choose a $\displaystyle (n+1)$-element subset from a $\displaystyle (n+1+r)$-element set, while on the left side there is the sum of ways we can select an $\displaystyle n$-element subset from a $\displaystyle n$-element, $\displaystyle (n+1)$-element,...,$\displaystyle (n+r)$-element set...

How can we prove that the numbers on both sides of the equation are actually identical?

I'd be really grateful for any help.

Thanks!