<s>[INST] Proving a combinatorics equality: $\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}$ How to prove the following? Should I use induction or something else?

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

Attempted start:
Basis step: ${\binom{1}{1}} = {\binom{2}{2}}$ true.
Where do I go from here?
 [/INST] Note that
$$\begin{align}
\binom ab+\binom a{b+1}&=\binom{a+1}{b+1}\\
\Rightarrow \binom ab&=\binom{a+1}{b+1}-\binom a{b+1}\end{align}$$
Hence 
$$\begin{align}
\binom ir&=\binom {i+1}{r+1}-\binom i{r+1}\\
\sum_{i=r}^n\binom ir&=\sum_{i=r}^n\binom {i+1}{r+1}-\sum_{i=r}^n\binom i{r+1}\\
&=\sum_{i=r+1}^{n+1}\binom {i}{r+1}-\sum_{i=r+1}^n\binom i{r+1}\\
&=\binom {n+1}{r+1}\qquad\blacksquare
\end{align}$$
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