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.

Proving by induction
inductive step
$$\begin{pmatrix}r\\r\end{pmatrix} + \begin{pmatrix}r+1\\r\end{pmatrix} + \dots + \begin{pmatrix}n\\r\end{pmatrix} =\begin{pmatrix}n\\r+1\end{pmatrix} + \begin{pmatrix}n\\r\end{pmatrix}$$ [as the identity holds for natural numbers less than n.]
We Know that, $$\begin{pmatrix}n\\r+1\end{pmatrix}+ \begin{pmatrix}n\\r\end{pmatrix} = \begin{pmatrix}n+1\\r+1\end{pmatrix}$$ see here
Hence proved.
