Algebraic proof for the following identity Give an algebraic proof that $\binom{n+1}{m+1} = \sum_{k=m}^{n} \binom{k}{m}$.
I've tried using Pascal's rule and looking for a telescopic sum, but I can't find one.
Any help is appreciated.


I've tried using Pascal's rule and looking for a telescopic sum,

How about
$$\binom{k}{m} + \binom{k}{m+1} = \binom{k+1}{m+1},$$
which transforms to
$$\binom{k}{m} = \binom{k+1}{m+1} - \binom{k}{m+1}$$
to nicely telescope?
