Induction proof concerning a sum of binomial coefficients: $\sum_{j=m}^n\binom{j}{m}=\binom{n+1}{m+1}$ I'm looking for a proof of this identity but where j=m not j=0
http://www.proofwiki.org/wiki/Sum_of_Binomial_Coefficients_over_Upper_Index
$$\sum_{j=m}^n\binom{j}{m}=\binom{n+1}{m+1}$$

It seems to me that you are looking for the proof of the identity: $$ \sum_{j=m}^n\binom{j}{m}=\binom{n+1}{m+1}$$
This is actually known as the Hockey-Stick Identity.You can find different methods of proving this in this page.
Please note that there are many applications of the hockey stick identity and all the combinatorial identities you may encounter, this is the one most worth remembering.  It 
makes a lot of apparently hard problems very easy. 
