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.

Assign numbers $1,2,3,\ldots,n,n+1$ to $n+1$ objects.  The number of ways to choose $r+1$ of them is $\dbinom{n+1}{r+1}$.
Either you choose the very last one and $r$ others bearing lower numbers (the number of ways to do that is $\dbinom n r$),
or you choose the one just before the last one and $r$ others bearing lower numbers (the number of ways to do that is $\dbinom{n-1}r$),
or you choose the one just before that one and $r$ others bearing lower numbers (the number of ways to do that is $\dbinom{n-2}r$),
or $\ldots\ldots$
