Pascal's Rule/Direct Proof

Theorem
Let $\displaystyle \binom n k$ be a binomial coefficient.

For positive integers $n, k$ with $1 \le k \le n$:
 * $\displaystyle \binom n {k-1} + \binom n k = \binom {n+1} k$

Proof
Let $n, k \in \N$ with $1 \le k \le n$.