The hockey-stick identity

Induction on $n$. Base $n=r$: $\binom{r}{r}=\binom{r+1}{r+1}=1$.

Step. Suppose $\sum_{k=r}^{n}\binom{k}{r}=\binom{n+1}{r+1}$. Then
$$\sum_{k=r}^{n+1}\binom{k}{r}=\binom{n+1}{r+1}+\binom{n+1}{r}=\binom{n+2}{r+1}$$
by Pascal.

Conclusion. The identity holds.

Name. Plotted on Pascal's triangle, the sum follows a hockey stick shape.