Rising Sum of Binomial Coefficients

Theorem
Let $m, n \in \Z_{\ge 0}$ be positive integers.

Then:
 * $\displaystyle \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1} = \binom {n + m + 1} m$

where $\dbinom n k$ denotes a binomial coefficient.

That is:
 * $\dbinom n n + \dbinom {n + 1} n + \dbinom {n + 2} n + \cdots + \dbinom {n + m} n = \dbinom {n + m + 1} {n + 1} = \dbinom {n + m + 1} m$

Marginal cases
Just to make sure, it is worth checking the marginal cases: