Proving binomial coefficients identity: $\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}$ 
Let $n$ and $r$ be positive integers with $n \ge r$. Prove that:
$$\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}.$$

Tried proving it by induction but got stuck. Any help with proving it by induction or any other proof technique is appreciated.

Let $E = \{1,2, \dots , n+1\}$. The number $\binom{n+1}{r+1}$ is the number of subsets $A$ of $E$ with $r + 1$ elements. 
Classify these subsets $A$ according to their largest element $b$, which can be any number among $r + 1$, $r + 2$, ..., $n + 1$. The number of $(r+1)$-element subsets of $E$ with largest element $b$ is the same as the number of $r$-element subsets of $\{1, 2, \dots, b-1\}$, which is $\binom{b-1}{r}$. 
Now $b$ can be any number $r + 1, r + 2, \ldots, n + 1$, and there are $\binom{r}{r}$, $\binom{r + 1}{r}, \ldots, \binom{n}{r}$ possible subsets $A$ in each case. This proves the desired equality.
