\begin{align*} \binom{n}{r} + \binom{n}{r+1} & = \frac{n!}{r!(n-r)!} + \frac{n!}{(r+1)!(n-(r+1))!}\\ & = (r+1) \frac{n!}{(r+1)!(n-r)!} \\ + (n-r) \frac{n!}{(r+1)!(n-r))!}\\ & = \frac{(r+1) n! + (n-r)n!}{(r+1)!(n-r)!}\\ & = \frac{(n+1) n!}{(r+1)!((n+1)-(r+1))!}\\ & = \binom{n+1}{r+1} \end{align*}